id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-6200 | Pointwise Addition on Rational-Valued Functions is Associative | Let $f, g, h: S \to \Q$ be rational-valued functions.
Let $f + g: S \to \Q$ denote the pointwise sum of $f$ and $g$.
Then:
:$\paren {f + g} + h = f + \paren {g + h}$ | {{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\paren {\paren {f + g} + h} } x
| r = \paren {\map f x + \map g x} + \map h x
| c = {{Defof|Pointwise Addition of Rational-Valued Functions}}
}}
{{eqn | r = \map f x + \paren {\map g x + \map h x}
| c = Rational Addition is Associative
}}
{{e... | Let $f, g, h: S \to \Q$ be [[Definition:Rational-Valued Function|rational-valued functions]].
Let $f + g: S \to \Q$ denote the [[Definition:Pointwise Addition of Rational-Valued Functions|pointwise sum of $f$ and $g$]].
Then:
:$\paren {f + g} + h = f + \paren {g + h}$ | {{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\paren {\paren {f + g} + h} } x
| r = \paren {\map f x + \map g x} + \map h x
| c = {{Defof|Pointwise Addition of Rational-Valued Functions}}
}}
{{eqn | r = \map f x + \paren {\map g x + \map h x}
| c = [[Rational Addition is Associative]]
}}... | Pointwise Addition on Rational-Valued Functions is Associative | https://proofwiki.org/wiki/Pointwise_Addition_on_Rational-Valued_Functions_is_Associative | https://proofwiki.org/wiki/Pointwise_Addition_on_Rational-Valued_Functions_is_Associative | [
"Pointwise Addition is Associative",
"Rational Addition"
] | [
"Definition:Rational-Valued Function",
"Definition:Pointwise Addition of Rational-Valued Functions"
] | [
"Rational Addition is Associative",
"Category:Pointwise Addition is Associative",
"Category:Rational Addition"
] |
proofwiki-6201 | Pointwise Multiplication on Integer-Valued Functions is Associative | Let $f, g, h: S \to \Z$ be integer-valued functions.
Let $f \times g: S \to \Z$ denote the pointwise product of $f$ and $g$.
Then:
:$\paren {f \times g} \times h = f \times \paren {g \times h}$ | {{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\paren {\paren {f \times g} \times h} } x
| r = \paren {\map f x \times \map g x} \times \map h x
| c = {{Defof|Pointwise Multiplication of Integer-Valued Functions}}
}}
{{eqn | r = \map f x \times \paren {\map g x \times \map h x}
| c = Inte... | Let $f, g, h: S \to \Z$ be [[Definition:Integer-Valued Function|integer-valued functions]].
Let $f \times g: S \to \Z$ denote the [[Definition:Pointwise Multiplication of Integer-Valued Functions|pointwise product of $f$ and $g$]].
Then:
:$\paren {f \times g} \times h = f \times \paren {g \times h}$ | {{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\paren {\paren {f \times g} \times h} } x
| r = \paren {\map f x \times \map g x} \times \map h x
| c = {{Defof|Pointwise Multiplication of Integer-Valued Functions}}
}}
{{eqn | r = \map f x \times \paren {\map g x \times \map h x}
| c = [[In... | Pointwise Multiplication on Integer-Valued Functions is Associative | https://proofwiki.org/wiki/Pointwise_Multiplication_on_Integer-Valued_Functions_is_Associative | https://proofwiki.org/wiki/Pointwise_Multiplication_on_Integer-Valued_Functions_is_Associative | [
"Pointwise Multiplication is Associative",
"Integer Multiplication"
] | [
"Definition:Integer-Valued Function",
"Definition:Pointwise Multiplication of Integer-Valued Functions"
] | [
"Integer Multiplication is Associative",
"Category:Pointwise Multiplication is Associative",
"Category:Integer Multiplication"
] |
proofwiki-6202 | Pointwise Multiplication on Complex-Valued Functions is Associative | Let $f, g, h: S \to \C$ be complex-valued functions.
Let $f \times g: S \to \C$ denote the pointwise product of $f$ and $g$.
Then:
:$\paren {f \times g} \times h = f \times \paren {g \times h}$ | {{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\paren {\paren {f \times g} \times h} } x
| r = \paren {\map f x \times \map g x} \times \map h c
| c = {{Defof|Pointwise Multiplication of Complex-Valued Functions}}
}}
{{eqn | r = \map f x \times \paren {\map g x \times \map h x}
| c = Comp... | Let $f, g, h: S \to \C$ be [[Definition:Complex-Valued Function|complex-valued functions]].
Let $f \times g: S \to \C$ denote the [[Definition:Pointwise Multiplication of Complex-Valued Functions|pointwise product of $f$ and $g$]].
Then:
:$\paren {f \times g} \times h = f \times \paren {g \times h}$ | {{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\paren {\paren {f \times g} \times h} } x
| r = \paren {\map f x \times \map g x} \times \map h c
| c = {{Defof|Pointwise Multiplication of Complex-Valued Functions}}
}}
{{eqn | r = \map f x \times \paren {\map g x \times \map h x}
| c = [[Co... | Pointwise Multiplication on Complex-Valued Functions is Associative | https://proofwiki.org/wiki/Pointwise_Multiplication_on_Complex-Valued_Functions_is_Associative | https://proofwiki.org/wiki/Pointwise_Multiplication_on_Complex-Valued_Functions_is_Associative | [
"Pointwise Multiplication is Associative",
"Complex Multiplication"
] | [
"Definition:Complex-Valued Function",
"Definition:Pointwise Multiplication of Complex-Valued Functions"
] | [
"Complex Multiplication is Associative",
"Category:Pointwise Multiplication is Associative",
"Category:Complex Multiplication"
] |
proofwiki-6203 | Pointwise Multiplication on Real-Valued Functions is Associative | Let $f, g, h: S \to \R$ be real-valued functions.
Let $f \times g: S \to \R$ denote the pointwise product of $f$ and $g$.
Then:
:$\paren {f \times g} \times h = f \times \paren {g \times h}$ | {{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\paren {\paren {f \times g} \times h} } x
| r = \paren {\map f x \times \map g x} \times \map h x
| c = {{Defof|Pointwise Multiplication of Real-Valued Functions}}
}}
{{eqn | r = \map f x \times \paren {\map g x \times \map h x}
| c = Real Mu... | Let $f, g, h: S \to \R$ be [[Definition:Real-Valued Function|real-valued functions]].
Let $f \times g: S \to \R$ denote the [[Definition:Pointwise Multiplication of Real-Valued Functions|pointwise product of $f$ and $g$]].
Then:
:$\paren {f \times g} \times h = f \times \paren {g \times h}$ | {{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\paren {\paren {f \times g} \times h} } x
| r = \paren {\map f x \times \map g x} \times \map h x
| c = {{Defof|Pointwise Multiplication of Real-Valued Functions}}
}}
{{eqn | r = \map f x \times \paren {\map g x \times \map h x}
| c = [[Real ... | Pointwise Multiplication on Real-Valued Functions is Associative | https://proofwiki.org/wiki/Pointwise_Multiplication_on_Real-Valued_Functions_is_Associative | https://proofwiki.org/wiki/Pointwise_Multiplication_on_Real-Valued_Functions_is_Associative | [
"Pointwise Multiplication is Associative",
"Real Multiplication"
] | [
"Definition:Real-Valued Function",
"Definition:Pointwise Multiplication of Real-Valued Functions"
] | [
"Real Multiplication is Associative",
"Category:Pointwise Multiplication is Associative",
"Category:Real Multiplication"
] |
proofwiki-6204 | Pointwise Multiplication on Rational-Valued Functions is Associative | Let $f, g, h: S \to \Q$ be rational-valued functions.
Let $f \times g: S \to \Q$ denote the pointwise product of $f$ and $g$.
Then:
:$\paren {f \times g} \times h = f \times \paren {g \times h}$ | {{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\paren {\paren {f \times g} \times h} } x
| r = \paren {\map f x \times \map g x} \times \map h x
| c = {{Defof|Pointwise Multiplication of Rational-Valued Functions}}
}}
{{eqn | r = \map f x \times \paren {\map g x \times \map h x}
| c = Rat... | Let $f, g, h: S \to \Q$ be [[Definition:Rational-Valued Function|rational-valued functions]].
Let $f \times g: S \to \Q$ denote the [[Definition:Pointwise Multiplication of Rational-Valued Functions|pointwise product of $f$ and $g$]].
Then:
:$\paren {f \times g} \times h = f \times \paren {g \times h}$ | {{begin-eqn}}
{{eqn | q = \forall x \in S
| l = \map {\paren {\paren {f \times g} \times h} } x
| r = \paren {\map f x \times \map g x} \times \map h x
| c = {{Defof|Pointwise Multiplication of Rational-Valued Functions}}
}}
{{eqn | r = \map f x \times \paren {\map g x \times \map h x}
| c = [[R... | Pointwise Multiplication on Rational-Valued Functions is Associative | https://proofwiki.org/wiki/Pointwise_Multiplication_on_Rational-Valued_Functions_is_Associative | https://proofwiki.org/wiki/Pointwise_Multiplication_on_Rational-Valued_Functions_is_Associative | [
"Pointwise Multiplication is Associative",
"Rational Multiplication"
] | [
"Definition:Rational-Valued Function",
"Definition:Pointwise Multiplication of Rational-Valued Functions"
] | [
"Rational Multiplication is Associative",
"Category:Pointwise Multiplication is Associative",
"Category:Rational Multiplication"
] |
proofwiki-6205 | Set Equation: Union | Let $A$ and $B$ be sets.
Consider the set equation:
:$A \cup X = B$
The solution set of this is given by:
:$X = \begin {cases} \O & : A \nsubseteq B \\ \set {\paren {B \setminus A} \cup Y: Y \subseteq A} & : \text {otherwise} \end {cases}$ | In the first case $A$ is a not a subset of $B$.
So there exists an $x \in A$ such that $x \notin B$.
By the definition of union:
:$\forall x: x \in A \implies x \in A \cup X$
Hence the solution set is empty.
In the second case, suppose $A$ is a subset of $B$.
{{WIP}}
In particular, suppose $A$ is a subset of $B$, and $... | Let $A$ and $B$ be [[Definition:Set|sets]].
Consider the [[Definition:Set Equation|set equation]]:
:$A \cup X = B$
The solution set of this is given by:
:$X = \begin {cases} \O & : A \nsubseteq B \\ \set {\paren {B \setminus A} \cup Y: Y \subseteq A} & : \text {otherwise} \end {cases}$ | In the first case $A$ is a not a subset of $B$.
So there exists an $x \in A$ such that $x \notin B$.
By the definition of [[Definition:Set Union|union]]:
:$\forall x: x \in A \implies x \in A \cup X$
Hence the solution set is [[Definition:Empty Set|empty]].
In the second case, suppose $A$ is a subset of $B$.
{... | Set Equation: Union | https://proofwiki.org/wiki/Set_Equation:_Union | https://proofwiki.org/wiki/Set_Equation:_Union | [
"Set Theory",
"Set Union"
] | [
"Definition:Set",
"Definition:Set Equation"
] | [
"Definition:Set Union",
"Definition:Empty Set",
"Definition:Empty Set",
"Subset of Empty Set",
"Union of Empty Set",
"Empty Set is Subset of All Sets",
"Definition:Subset",
"Proof by Contraposition",
"Set is Subset of Itself",
"Definition:Existential Quantifier",
"Definition:Set Union",
"Set i... |
proofwiki-6206 | Set Equation: Intersection | Let $A$ and $B$ be sets.
Consider the set equation:
:$A \cap X = B$
The solution set of this is:
:$\O$ if $B \nsubseteq A$
:$\set {B \cup Y: A \nsubseteq Y}$ otherwise. | {{ProofWanted}}
Category:Set Theory
Category:Set Intersection
p9k3ot62zxrq1wcpmbaq5deehrvs41t | Let $A$ and $B$ be [[Definition:Set|sets]].
Consider the [[Definition:Set Equation|set equation]]:
:$A \cap X = B$
The solution set of this is:
:$\O$ if $B \nsubseteq A$
:$\set {B \cup Y: A \nsubseteq Y}$ otherwise. | {{ProofWanted}}
[[Category:Set Theory]]
[[Category:Set Intersection]]
p9k3ot62zxrq1wcpmbaq5deehrvs41t | Set Equation: Intersection | https://proofwiki.org/wiki/Set_Equation:_Intersection | https://proofwiki.org/wiki/Set_Equation:_Intersection | [
"Set Theory",
"Set Intersection"
] | [
"Definition:Set",
"Definition:Set Equation"
] | [
"Category:Set Theory",
"Category:Set Intersection"
] |
proofwiki-6207 | Non-Abelian Group has Order Greater than 4 | Let $\struct {G, \circ}$ be a non-abelian group.
Then the order of $\struct {G, \circ}$ is greater than $4$. | Let $\left({G, \circ}\right)$ be a non-abelian group whose identity is $e$.
For a group $\left({G, \circ}\right)$ to be non-abelian, we require:
: $\exists x, y \in G: x \circ y \ne y \circ x$
Suppose $x \circ y \in \left\{ {x, y, e}\right\}$.
: $x \circ y = e \implies y \circ x = e$
and $\left({G, \circ}\right)$ is ab... | Let $\struct {G, \circ}$ be a non-[[Definition:Abelian Group|abelian group]].
Then the [[Definition:Order of Structure|order]] of $\struct {G, \circ}$ is greater than $4$. | Let $\left({G, \circ}\right)$ be a non-[[Definition:Abelian Group|abelian group]] whose [[Definition:Identity Element|identity]] is $e$.
For a group $\left({G, \circ}\right)$ to be non-[[Definition:Abelian Group|abelian]], we require:
: $\exists x, y \in G: x \circ y \ne y \circ x$
Suppose $x \circ y \in \left\{ {x, ... | Non-Abelian Group has Order Greater than 4/Proof 1 | https://proofwiki.org/wiki/Non-Abelian_Group_has_Order_Greater_than_4 | https://proofwiki.org/wiki/Non-Abelian_Group_has_Order_Greater_than_4/Proof_1 | [
"Abelian Groups",
"Non-Abelian Group has Order Greater than 4"
] | [
"Definition:Abelian Group",
"Definition:Order of Structure"
] | [
"Definition:Abelian Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Abelian Group",
"Definition:Abelian Group",
"Definition:Abelian Group",
"Definition:Element",
"Definition:Abelian Group",
"Definition:Element"
] |
proofwiki-6208 | Non-Abelian Group has Order Greater than 4 | Let $\struct {G, \circ}$ be a non-abelian group.
Then the order of $\struct {G, \circ}$ is greater than $4$. | It follows from Trivial Group is Cyclic Group and Prime Group is Cyclic that groups of order less than $4$ are cyclic.
Therefore, by Cyclic Group is Abelian, all groups of order less than $4$ are abelian.
Let $G$ have order $4$.
From Order of Element Divides Order of Finite Group, every element of $G$ has order that di... | Let $\struct {G, \circ}$ be a non-[[Definition:Abelian Group|abelian group]].
Then the [[Definition:Order of Structure|order]] of $\struct {G, \circ}$ is greater than $4$. | It follows from [[Trivial Group is Cyclic Group]] and [[Prime Group is Cyclic]] that [[Definition:Group|groups]] of [[Definition:Order of Structure|order]] less than $4$ are [[Definition:Cyclic Group|cyclic]].
Therefore, by [[Cyclic Group is Abelian]], all [[Definition:Group|groups]] of [[Definition:Order of Structure... | Non-Abelian Group has Order Greater than 4/Proof 2 | https://proofwiki.org/wiki/Non-Abelian_Group_has_Order_Greater_than_4 | https://proofwiki.org/wiki/Non-Abelian_Group_has_Order_Greater_than_4/Proof_2 | [
"Abelian Groups",
"Non-Abelian Group has Order Greater than 4"
] | [
"Definition:Abelian Group",
"Definition:Order of Structure"
] | [
"Trivial Group is Cyclic Group",
"Prime Group is Cyclic",
"Definition:Group",
"Definition:Order of Structure",
"Definition:Cyclic Group",
"Cyclic Group is Abelian",
"Definition:Group",
"Definition:Order of Structure",
"Definition:Abelian Group",
"Definition:Order of Structure",
"Order of Element... |
proofwiki-6209 | Bayes' Theorem/General Result | Let $\set {B_1, B_2, \ldots}$ be a partition of the event space $\Sigma$.
Then, for any $B_i$ in the partition:
:$\condprob {B_i} A = \dfrac {\condprob A {B_i} \map \Pr {B_i} } {\map \Pr A} = \dfrac {\condprob A {B_i} \map \Pr {B_i} } {\sum_j \condprob A {B_j} \map \Pr {B_j} }$
where $\ds \sum_j$ denotes the sum over $... | Follows directly from the Total Probability Theorem:
:$\ds \map \Pr A = \sum_i \condprob A {B_i} \map \Pr {B_i}$
and Bayes' Theorem:
:$\condprob {B_i} A = \dfrac {\condprob A {B_i} \map \Pr {B_i} } {\map \Pr A}$
{{qed}} | Let $\set {B_1, B_2, \ldots}$ be a [[Definition:Partition (Probability Theory)|partition]] of the [[Definition:Event Space|event space]] $\Sigma$.
Then, for any $B_i$ in the [[Definition:Partition (Probability Theory)|partition]]:
:$\condprob {B_i} A = \dfrac {\condprob A {B_i} \map \Pr {B_i} } {\map \Pr A} = \dfrac... | Follows directly from the [[Total Probability Theorem]]:
:$\ds \map \Pr A = \sum_i \condprob A {B_i} \map \Pr {B_i}$
and [[Bayes' Theorem]]:
:$\condprob {B_i} A = \dfrac {\condprob A {B_i} \map \Pr {B_i} } {\map \Pr A}$
{{qed}} | Bayes' Theorem/General Result | https://proofwiki.org/wiki/Bayes'_Theorem/General_Result | https://proofwiki.org/wiki/Bayes'_Theorem/General_Result | [
"Bayes' Theorem"
] | [
"Definition:Partition (Probability Theory)",
"Definition:Event Space",
"Definition:Partition (Probability Theory)",
"Definition:Summation"
] | [
"Total Probability Theorem",
"Bayes' Theorem"
] |
proofwiki-6210 | Supremum is Coproduct in Order Category | Let $\mathbf P$ be an order category with ordering $\preceq$.
Let $p, q \in P_0$, and suppose they have some supremum $r = \sup \left\{{p, q}\right\}$.
Then $r$ is the coproduct of $p$ and $q$ in $\mathbf P$. | Let $\mathbf P^{\text{op}}$ be the dual category of $\mathbf P$.
From Dual of Order Category, it is the order category corresponding to the dual ordering $\succeq$.
From Dual Pairs (Order Theory), it follows that in $\mathbf P^{\text{op}}$:
:$r = \inf \left\{{p, q}\right\}$
where $\inf$ denotes infimum.
By Infimum is P... | Let $\mathbf P$ be an [[Definition:Order Category|order category]] with [[Definition:Ordering|ordering]] $\preceq$.
Let $p, q \in P_0$, and suppose they have some [[Definition:Supremum of Set|supremum]] $r = \sup \left\{{p, q}\right\}$.
Then $r$ is the [[Definition:Coproduct|coproduct]] of $p$ and $q$ in $\mathbf P$... | Let $\mathbf P^{\text{op}}$ be the [[Definition:Dual Category|dual category]] of $\mathbf P$.
From [[Dual of Order Category]], it is the [[Definition:Order Category|order category]] corresponding to the [[Definition:Dual Ordering|dual ordering]] $\succeq$.
From [[Dual Pairs (Order Theory)]], it follows that in $\mat... | Supremum is Coproduct in Order Category | https://proofwiki.org/wiki/Supremum_is_Coproduct_in_Order_Category | https://proofwiki.org/wiki/Supremum_is_Coproduct_in_Order_Category | [
"Coproducts",
"Order Categories"
] | [
"Definition:Order Category",
"Definition:Ordering",
"Definition:Supremum of Set",
"Definition:Coproduct"
] | [
"Definition:Dual Category",
"Dual of Order Category",
"Definition:Order Category",
"Definition:Dual Ordering",
"Dual Pairs (Order Theory)",
"Definition:Infimum of Set",
"Infimum is Product in Order Category",
"Definition:Product (Category Theory)/Binary Product",
"Dual Pairs (Category Theory)",
"D... |
proofwiki-6211 | Smooth Homotopy is an Equivalence Relation | Let $X$ and $Y$ be smooth manifolds.
Let $K \subseteq X$ be a (possibly empty) subset of $X$.
Let $\map {\CC^\infty} {X, Y}$ be the set of all smooth mappings from $X$ to $Y$.
Define a relation $\sim$ on $\map \CC {X, Y}$ by $f \sim g$ if $f$ and $g$ are smoothly homotopic relative to $K$.
Then $\sim$ is an equivalence... | We examine each condition for equivalence. | Let $X$ and $Y$ be [[Definition:Smooth Manifold|smooth manifolds]].
Let $K \subseteq X$ be a (possibly [[Definition:Empty Set|empty]]) [[Definition:Subset|subset]] of $X$.
Let $\map {\CC^\infty} {X, Y}$ be the [[Definition:Set|set]] of all [[Definition:Smooth Mapping (Manifolds)|smooth mappings]] from $X$ to $Y$.
De... | We examine each condition for [[Definition:Equivalence Relation|equivalence]]. | Smooth Homotopy is an Equivalence Relation | https://proofwiki.org/wiki/Smooth_Homotopy_is_an_Equivalence_Relation | https://proofwiki.org/wiki/Smooth_Homotopy_is_an_Equivalence_Relation | [
"Homotopy Theory",
"Examples of Equivalence Relations"
] | [
"Definition:Topological Manifold/Smooth Manifold",
"Definition:Empty Set",
"Definition:Subset",
"Definition:Set",
"Definition:Smooth Mapping (Manifolds)",
"Definition:Relation",
"Definition:Smooth Homotopy",
"Definition:Equivalence Relation"
] | [
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-6212 | Real Function is Continuous at Isolated Point | Let $A \subseteq \R$ be any subset of the real numbers.
Let $f: A \to \R$ be a real function.
Let $x \in A$ be an isolated point of $A$.
Then $f$ is continuous at $x$. | By definition of isolation point:
:$\exists \delta \in \R_{> 0}: \openint {x - \delta} {x + \delta} \cap A = \set x$
Pick any $\epsilon > 0$.
We have that for any $z \in \openint {x - \delta} {x + \delta} \cap A = \set x$:
:$\size {\map f z - \map f x} = \size {\map f x - \map f x} = 0 < \epsilon$.
This satisfies the c... | Let $A \subseteq \R$ be any [[Definition:Subset|subset]] of the [[Definition:Real Number|real numbers]].
Let $f: A \to \R$ be a [[Definition:Real Function|real function]].
Let $x \in A$ be an [[Definition:Isolated Point (Real Analysis)|isolated point]] of $A$.
Then $f$ is [[Definition:Continuous Real Function at Po... | By definition of [[Definition:Isolated Point (Real Analysis)|isolation point]]:
:$\exists \delta \in \R_{> 0}: \openint {x - \delta} {x + \delta} \cap A = \set x$
Pick any $\epsilon > 0$.
We have that for any $z \in \openint {x - \delta} {x + \delta} \cap A = \set x$:
:$\size {\map f z - \map f x} = \size {\map f ... | Real Function is Continuous at Isolated Point | https://proofwiki.org/wiki/Real_Function_is_Continuous_at_Isolated_Point | https://proofwiki.org/wiki/Real_Function_is_Continuous_at_Isolated_Point | [
"Continuity",
"Real Functions"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Real Function",
"Definition:Isolated Point (Real Analysis)",
"Definition:Continuous Real Function/Point"
] | [
"Definition:Isolated Point (Real Analysis)",
"Definition:Continuous Real Function/Point",
"Definition:Limit of Real Function",
"Category:Continuity",
"Category:Real Functions"
] |
proofwiki-6213 | Metric Space Continuity by Epsilon-Delta | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$.
Let $a \in A_1$ be a point in $A_1$.
Then the following definitions of continuity of $f$ at $a$ with respect to $d_1$ and $d_2$ are equivalent: | === Definition by Limits implies $\epsilon$-$\delta$ Definition ===
Suppose that $f$ is $\tuple {d_1, d_2}$-continuous at $a$ in the sense that:
:$(1): \quad$ The limit of $\map f x$ as $x \to a$ exists
:$(2): \quad \ds \lim_{x \mathop \to a} \map f x = \map f a$
Let $\ds \lim_{x \mathop \to a} \map f x$.
Then by the $... | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $f: A_1 \to A_2$ be a [[Definition:Mapping|mapping]] from $A_1$ to $A_2$.
Let $a \in A_1$ be a point in $A_1$.
Then the following definitions of [[Definition:Continuous at Point of Metric Space|continuity... | === Definition by Limits implies $\epsilon$-$\delta$ Definition ===
Suppose that [[Definition:Continuous Mapping (Metric Space)/Point/Definition 1|$f$ is $\tuple {d_1, d_2}$-continuous at $a$]] in the sense that:
:$(1): \quad$ The [[Definition:Limit of Mapping between Metric Spaces|limit]] of $\map f x$ as $x \to a$ e... | Metric Space Continuity by Epsilon-Delta | https://proofwiki.org/wiki/Metric_Space_Continuity_by_Epsilon-Delta | https://proofwiki.org/wiki/Metric_Space_Continuity_by_Epsilon-Delta | [
"Continuous Mappings on Metric Spaces",
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Mapping",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Logical Equivalence"
] | [
"Definition:Continuous Mapping (Metric Space)/Point/Definition 1",
"Definition:Limit of Mapping between Metric Spaces",
"Definition:Limit of Mapping between Metric Spaces/Epsilon-Delta Condition",
"Definition:Continuous Mapping (Metric Space)/Point/Definition 2",
"Definition:Continuous Mapping (Metric Space... |
proofwiki-6214 | Equalizer is Monomorphism | Let $\mathbf C$ be a metacategory.
Let $e: E \to C$ be the equalizer of two morphisms $f, g: C \to D$.
Then $e$ is a monomorphism. | Suppose that for morphisms $x,y: Z \to E$, it holds that:
:$e \circ y = e \circ x$
Putting $z = e \circ x$, the following commutative diagram applies:
$\quad\quad \begin{xy}\xymatrix{
E
\ar[r]^*{e}
&
C
\ar[r]<2pt>^*{f}
\ar[r]<-2pt>_*{g}
&
D
\\
Z
\ar[u]<2pt>^*{x}
\ar[u]<-2pt>_*{y}
\ar[ur]_*{z}
}\end{xy}$
It follows t... | Let $\mathbf C$ be a [[Definition:Metacategory|metacategory]].
Let $e: E \to C$ be the [[Definition:Equalizer|equalizer]] of two [[Definition:Morphism|morphisms]] $f, g: C \to D$.
Then $e$ is a [[Definition:Monomorphism (Category Theory)|monomorphism]]. | Suppose that for [[Definition:Morphism|morphisms]] $x,y: Z \to E$, it holds that:
:$e \circ y = e \circ x$
Putting $z = e \circ x$, the following [[Definition:Commutative Diagram|commutative diagram]] applies:
$\quad\quad \begin{xy}\xymatrix{
E
\ar[r]^*{e}
&
C
\ar[r]<2pt>^*{f}
\ar[r]<-2pt>_*{g}
&
D
\\
Z
\ar[u]<2p... | Equalizer is Monomorphism | https://proofwiki.org/wiki/Equalizer_is_Monomorphism | https://proofwiki.org/wiki/Equalizer_is_Monomorphism | [
"Monomorphisms (Category Theory)"
] | [
"Definition:Metacategory",
"Definition:Equalizer",
"Definition:Morphism",
"Definition:Monomorphism (Category Theory)"
] | [
"Definition:Morphism",
"Definition:Commutative Diagram",
"Definition:Equalizer",
"Definition:Monomorphism (Category Theory)"
] |
proofwiki-6215 | Metric Space Continuity by Open Ball | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$.
Let $a \in A_1$ be a point in $A_1$.
{{TFAE|def = Continuous at Point of Metric Space|view = continuity of $f$ at $a$ with respect to $d_1$ and $d_2$}} | === $\epsilon$-$\delta$ Definition implies $\epsilon$-Ball Definition ===
Suppose that $f$ is $\tuple {d_1, d_2}$-continuous at $a$ in the sense that:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in A_1: \map {d_1} {x, a} < \delta \implies \map {d_2} {\map f x, \map f a} < \epsilon$
where $\R_... | Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $f: A_1 \to A_2$ be a [[Definition:Mapping|mapping]] from $A_1$ to $A_2$.
Let $a \in A_1$ be a point in $A_1$.
{{TFAE|def = Continuous at Point of Metric Space|view = continuity of $f$ at $a$ with respect... | === $\epsilon$-$\delta$ Definition implies $\epsilon$-Ball Definition ===
Suppose that [[Definition:Continuous Mapping (Metric Space)/Point/Definition 1|$f$ is $\tuple {d_1, d_2}$-continuous at $a$]] in the sense that:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in A_1: \map {d_1} {x, a} < \... | Metric Space Continuity by Open Ball | https://proofwiki.org/wiki/Metric_Space_Continuity_by_Open_Ball | https://proofwiki.org/wiki/Metric_Space_Continuity_by_Open_Ball | [
"Open Balls",
"Continuous Mappings on Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Mapping"
] | [
"Definition:Continuous Mapping (Metric Space)/Point/Definition 1",
"Definition:Set",
"Definition:Strictly Positive/Real Number",
"Definition:By Hypothesis",
"Definition:By Hypothesis",
"Definition:Continuous Mapping (Metric Space)/Point/Definition 1",
"Definition:By Hypothesis"
] |
proofwiki-6216 | Mapping from Standard Discrete Metric on Real Number Line is Continuous | Let $\R$ be the real number line.
Let $\struct {\R, d_1}$ be the metric space such that $d_1$ be the Euclidean metric on $\R$.
Let $\struct {\R, d_2}$ be the metric space such that $d_2$ be the standard discrete metric on $\R$.
Let $f: \tuple {\R, d_2} \to \tuple {\R, d_1}$ be a real function.
Then $f$ is $\tuple {d_2,... | Let $\epsilon \in \R: \epsilon > 0$.
Let $\delta = 1$.
Let $x \in \R$.
Let $y \in \R$ such that $\map {d_2} {x, y} < \delta$.
That is, $\map {d_2} {x, y} < 1$.
By the definition of the standard discrete metric on $\R$, that would mean that $\map {d_2} {x, y} = 0$ and so $x = y$.
Thus $\map f x = \map f y$.
By definitio... | Let $\R$ be the [[Definition:Real Number Line|real number line]].
Let $\struct {\R, d_1}$ be the [[Definition:Metric Space|metric space]] such that $d_1$ be the [[Definition:Euclidean Metric on Real Number Line|Euclidean metric]] on $\R$.
Let $\struct {\R, d_2}$ be the [[Definition:Metric Space|metric space]] such th... | Let $\epsilon \in \R: \epsilon > 0$.
Let $\delta = 1$.
Let $x \in \R$.
Let $y \in \R$ such that $\map {d_2} {x, y} < \delta$.
That is, $\map {d_2} {x, y} < 1$.
By the definition of the [[Definition:Standard Discrete Metric|standard discrete metric]] on $\R$, that would mean that $\map {d_2} {x, y} = 0$ and so $x ... | Mapping from Standard Discrete Metric on Real Number Line is Continuous | https://proofwiki.org/wiki/Mapping_from_Standard_Discrete_Metric_on_Real_Number_Line_is_Continuous | https://proofwiki.org/wiki/Mapping_from_Standard_Discrete_Metric_on_Real_Number_Line_is_Continuous | [
"Standard Discrete Metric",
"Continuous Mappings"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Metric Space",
"Definition:Euclidean Metric/Real Number Line",
"Definition:Metric Space",
"Definition:Standard Discrete Metric",
"Definition:Real Function",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Standard Discrete Metric",
"Definition:Metric Space/Metric",
"Definition:Continuous Mapping (Metric Space)/Point",
"Definition:Continuous Mapping (Metric Space)/Space"
] |
proofwiki-6217 | Properties of Affine Spaces | Let $\EE$ be an affine space with difference space $V$.
Let $0$ denote the zero element of $V$.
Then the following hold for all $p,q,r \in \EE$ and all $u, v \in V$:
:$(1): \quad p - p = 0$
:$(2): \quad p + 0 = p$
:$(3): \quad p + u = p + v \iff u = v$
:$(4): \quad q - p = r - p \iff q = r$ | === $(1): \quad p - p = 0$ ===
We have:
{{begin-eqn}}
{{eqn | l = \paren {p - p} + \paren {q - p}
| r = \paren {p + \paren {q - p} } - p
}}
{{eqn | r = q - p
}}
{{end-eqn}}
From Zero Element is Unique:
:$p - p = 0$
{{qed|lemma}} | Let $\EE$ be an [[Definition:Affine Space|affine space]] with [[Definition:Difference Space|difference space]] $V$.
Let $0$ denote the [[Definition:Zero Element|zero element]] of $V$.
Then the following hold for all $p,q,r \in \EE$ and all $u, v \in V$:
:$(1): \quad p - p = 0$
:$(2): \quad p + 0 = p$
:$(3): \quad p ... | === $(1): \quad p - p = 0$ ===
We have:
{{begin-eqn}}
{{eqn | l = \paren {p - p} + \paren {q - p}
| r = \paren {p + \paren {q - p} } - p
}}
{{eqn | r = q - p
}}
{{end-eqn}}
From [[Zero Element is Unique]]:
:$p - p = 0$
{{qed|lemma}} | Properties of Affine Spaces | https://proofwiki.org/wiki/Properties_of_Affine_Spaces | https://proofwiki.org/wiki/Properties_of_Affine_Spaces | [
"Affine Geometry"
] | [
"Definition:Affine Space",
"Definition:Tangent Space (Affine Geometry)",
"Definition:Zero Element"
] | [
"Zero Element is Unique"
] |
proofwiki-6218 | Surjection from Natural Numbers iff Right Inverse | Let $S$ be a set.
Let $f: \N \to S$ be a mapping, where $\N$ denotes the set of natural numbers.
Then $f$ is a surjection {{iff}} $f$ admits a right inverse. | === Necessary Condition ===
Suppose that $g: S \to \N$ is a right inverse of $f$.
That is, let $f \circ g = I_S$, the identity mapping on $S$.
We have that $I_S$ is a surjection.
By Surjection if Composite is Surjection, it follows that $f$ is a surjection.
{{qed|lemma}} | Let $S$ be a [[Definition:Set|set]].
Let $f: \N \to S$ be a [[Definition:Mapping|mapping]], where $\N$ denotes the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]].
Then $f$ is a [[Definition:Surjection|surjection]] {{iff}} $f$ [[Definition:Existential Quantifier|admits]] a [[Definition:Right... | === Necessary Condition ===
Suppose that $g: S \to \N$ is a [[Definition:Right Inverse Mapping|right inverse]] of $f$.
That is, let $f \circ g = I_S$, the [[Definition:Identity Mapping|identity mapping]] on $S$.
We have that [[Identity Mapping is Surjection|$I_S$ is a surjection]].
By [[Surjection if Composite is ... | Surjection from Natural Numbers iff Right Inverse | https://proofwiki.org/wiki/Surjection_from_Natural_Numbers_iff_Right_Inverse | https://proofwiki.org/wiki/Surjection_from_Natural_Numbers_iff_Right_Inverse | [
"Surjections",
"Natural Numbers"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Surjection",
"Definition:Existential Quantifier",
"Definition:Right Inverse Mapping"
] | [
"Definition:Right Inverse Mapping",
"Definition:Identity Mapping",
"Identity Mapping is Surjection",
"Surjection if Composite is Surjection",
"Definition:Surjection",
"Definition:Surjection",
"Definition:Surjection",
"Definition:Right Inverse Mapping"
] |
proofwiki-6219 | Sum of Squared Deviations from Mean | Let $S = \set {x_1, x_2, \ldots, x_n}$ be a set of real numbers.
Let $\overline x$ denote the arithmetic mean of $S$.
Then:
:$\ds \sum_{i \mathop = 1}^n \paren {x_i - \overline x}^2 = \sum_{i \mathop = 1}^n \paren { {x_i}^2 - {\overline x}^2}$ | For brevity, let us write $\ds \sum$ for $\ds \sum_{i \mathop = 1}^n$.
Then:
{{begin-eqn}}
{{eqn | l = \sum \paren {x_i - \overline x}^2
| r = \sum \paren {x_i - \overline x} \paren {x_i - \overline x}
}}
{{eqn | r = \sum x_i \paren {x_i - \overline x} - \overline x \sum \paren {x_i - \overline x}
| c = Sum... | Let $S = \set {x_1, x_2, \ldots, x_n}$ be a [[Definition:Set|set]] of [[Definition:Real Number|real numbers]].
Let $\overline x$ denote the [[Definition:Arithmetic Mean|arithmetic mean]] of $S$.
Then:
:$\ds \sum_{i \mathop = 1}^n \paren {x_i - \overline x}^2 = \sum_{i \mathop = 1}^n \paren { {x_i}^2 - {\overline x... | For brevity, let us write $\ds \sum$ for $\ds \sum_{i \mathop = 1}^n$.
Then:
{{begin-eqn}}
{{eqn | l = \sum \paren {x_i - \overline x}^2
| r = \sum \paren {x_i - \overline x} \paren {x_i - \overline x}
}}
{{eqn | r = \sum x_i \paren {x_i - \overline x} - \overline x \sum \paren {x_i - \overline x}
| c = [... | Sum of Squared Deviations from Mean/Proof 1 | https://proofwiki.org/wiki/Sum_of_Squared_Deviations_from_Mean | https://proofwiki.org/wiki/Sum_of_Squared_Deviations_from_Mean/Proof_1 | [
"Arithmetic Mean",
"Sum of Squared Deviations from Mean"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Arithmetic Mean"
] | [
"Summation is Linear",
"Sum of Deviations from Mean",
"Sum of Deviations from Mean",
"Summation is Linear"
] |
proofwiki-6220 | Sum of Squared Deviations from Mean | Let $S = \set {x_1, x_2, \ldots, x_n}$ be a set of real numbers.
Let $\overline x$ denote the arithmetic mean of $S$.
Then:
:$\ds \sum_{i \mathop = 1}^n \paren {x_i - \overline x}^2 = \sum_{i \mathop = 1}^n \paren { {x_i}^2 - {\overline x}^2}$ | In this context, $x_1, x_2, \ldots, x_n$ are instances of a discrete random variable.
Hence the result Variance as Expectation of Square minus Square of Expectation can be applied:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
which means the same as this but in the language of probability theory.
{{qed}} | Let $S = \set {x_1, x_2, \ldots, x_n}$ be a [[Definition:Set|set]] of [[Definition:Real Number|real numbers]].
Let $\overline x$ denote the [[Definition:Arithmetic Mean|arithmetic mean]] of $S$.
Then:
:$\ds \sum_{i \mathop = 1}^n \paren {x_i - \overline x}^2 = \sum_{i \mathop = 1}^n \paren { {x_i}^2 - {\overline x... | In this context, $x_1, x_2, \ldots, x_n$ are instances of a [[Definition:Discrete Random Variable|discrete random variable]].
Hence the result [[Variance as Expectation of Square minus Square of Expectation]] can be applied:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
which means the same as this but in the lang... | Sum of Squared Deviations from Mean/Proof 2 | https://proofwiki.org/wiki/Sum_of_Squared_Deviations_from_Mean | https://proofwiki.org/wiki/Sum_of_Squared_Deviations_from_Mean/Proof_2 | [
"Arithmetic Mean",
"Sum of Squared Deviations from Mean"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Arithmetic Mean"
] | [
"Definition:Random Variable/Discrete",
"Variance as Expectation of Square minus Square of Expectation",
"Definition:Probability Theory"
] |
proofwiki-6221 | Surjection from Natural Numbers iff Countable | Let $S$ be a non-empty set.
Then $S$ is countable {{iff}} there exists a surjection $f: \N \to S$. | === Necessary Condition ===
Suppose that $f: \N \to S$ is a surjection.
By Surjection from Natural Numbers iff Right Inverse, $f$ admits a right inverse $g: S \to \N$.
We have that $g$ is an injection by Right Inverse Mapping is Injection.
Hence the result, by the definition of a countable set.
{{qed|lemma}} | Let $S$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|set]].
Then $S$ is [[Definition:Countable|countable]] {{iff}} [[Definition:Existential Quantifier|there exists]] a [[Definition:Surjection|surjection]] $f: \N \to S$. | === Necessary Condition ===
Suppose that $f: \N \to S$ is a [[Definition:Surjection|surjection]].
By [[Surjection from Natural Numbers iff Right Inverse]], $f$ [[Definition:Existential Quantifier|admits]] a [[Definition:Right Inverse Mapping|right inverse]] $g: S \to \N$.
We have that $g$ is an [[Definition:Injectio... | Surjection from Natural Numbers iff Countable | https://proofwiki.org/wiki/Surjection_from_Natural_Numbers_iff_Countable | https://proofwiki.org/wiki/Surjection_from_Natural_Numbers_iff_Countable | [
"Countable Sets",
"Natural Numbers",
"Surjections"
] | [
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Countable Set",
"Definition:Existential Quantifier",
"Definition:Surjection"
] | [
"Definition:Surjection",
"Surjection from Natural Numbers iff Right Inverse",
"Definition:Existential Quantifier",
"Definition:Right Inverse Mapping",
"Definition:Injection",
"Right Inverse Mapping is Injection",
"Definition:Countable Set",
"Definition:Countable Set",
"Definition:Injection",
"Defi... |
proofwiki-6222 | Sum of Identical Terms | Let $x$ be a number.
Let $n \in \N$ be a natural number such that $n \ge 1$.
Then:
:$\ds \sum_{i \mathop = 1}^n x = n x$
{{explain|Why limit this to $n \ge 1$? It also works for zero.}} | {{finish|this could be actually nontrivial; induction on $n$ seems easiest}}
{{expand|generalize to $x$ an element of a vector space, or for that matter, any abelian group}}
Category:Numbers
ibd586ciul8200zlipc2qntxpo013c4 | Let $x$ be a [[Definition:Number|number]].
Let $n \in \N$ be a [[Definition:Natural Number|natural number]] such that $n \ge 1$.
Then:
:$\ds \sum_{i \mathop = 1}^n x = n x$
{{explain|Why limit this to $n \ge 1$? It also works for zero.}} | {{finish|this could be actually nontrivial; induction on $n$ seems easiest}}
{{expand|generalize to $x$ an element of a vector space, or for that matter, any abelian group}}
[[Category:Numbers]]
ibd586ciul8200zlipc2qntxpo013c4 | Sum of Identical Terms | https://proofwiki.org/wiki/Sum_of_Identical_Terms | https://proofwiki.org/wiki/Sum_of_Identical_Terms | [
"Numbers"
] | [
"Definition:Number",
"Definition:Natural Numbers"
] | [
"Category:Numbers"
] |
proofwiki-6223 | Synthetic Sub-Basis and Analytic Sub-Basis are Compatible | Let $\struct {X, \tau}$ be a topological space.
Let $\SS \subseteq \powerset X$, where $\powerset X$ denotes the power set of $X$.
Then $\SS$ is an analytic sub-basis for $\tau$ {{iff}} $\tau$ is the topology on $X$ generated by the synthetic sub-basis $\SS$. | === Necessary Condition ===
Follows directly from the definitions of the generated topology and an analytic sub-basis.
{{qed|lemma}} | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\SS \subseteq \powerset X$, where $\powerset X$ denotes the [[Definition:Power Set|power set]] of $X$.
Then $\SS$ is an [[Definition:Analytic Sub-Basis|analytic sub-basis]] for $\tau$ {{iff}} $\tau$ is the [[Definition:Topology Gen... | === Necessary Condition ===
Follows directly from the definitions of the [[Definition:Topology Generated by Synthetic Sub-Basis/Definition 1|generated topology]] and an [[Definition:Analytic Sub-Basis|analytic sub-basis]].
{{qed|lemma}} | Synthetic Sub-Basis and Analytic Sub-Basis are Compatible | https://proofwiki.org/wiki/Synthetic_Sub-Basis_and_Analytic_Sub-Basis_are_Compatible | https://proofwiki.org/wiki/Synthetic_Sub-Basis_and_Analytic_Sub-Basis_are_Compatible | [
"Analytic Sub-Bases",
"Synthetic Sub-Bases"
] | [
"Definition:Topological Space",
"Definition:Power Set",
"Definition:Sub-Basis/Analytic Sub-Basis",
"Definition:Topology Generated by Synthetic Sub-Basis"
] | [
"Definition:Topology Generated by Synthetic Sub-Basis/Definition 1",
"Definition:Sub-Basis/Analytic Sub-Basis",
"Definition:Sub-Basis/Analytic Sub-Basis",
"Definition:Topology Generated by Synthetic Sub-Basis",
"Definition:Topology Generated by Synthetic Sub-Basis/Definition 1",
"Definition:Sub-Basis/Anal... |
proofwiki-6224 | Analytic Basis is Analytic Sub-Basis | Let $\struct {X, \tau}$ be a topological space.
Let $\BB \subseteq \tau$ be an analytic basis for $\tau$.
Then $\BB$ is an analytic sub-basis for $\tau$. | {{ProofWanted}}
Category:Topological Bases
ji3fvxw8xlmd0qk7wbmt0amhqe752sj | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\BB \subseteq \tau$ be an [[Definition:Analytic Basis|analytic basis]] for $\tau$.
Then $\BB$ is an [[Definition:Analytic Sub-Basis|analytic sub-basis]] for $\tau$. | {{ProofWanted}}
[[Category:Topological Bases]]
ji3fvxw8xlmd0qk7wbmt0amhqe752sj | Analytic Basis is Analytic Sub-Basis | https://proofwiki.org/wiki/Analytic_Basis_is_Analytic_Sub-Basis | https://proofwiki.org/wiki/Analytic_Basis_is_Analytic_Sub-Basis | [
"Topological Bases"
] | [
"Definition:Topological Space",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Sub-Basis/Analytic Sub-Basis"
] | [
"Category:Topological Bases"
] |
proofwiki-6225 | Continuity Test using Sub-Basis | Let $\struct {X_1, \tau_1}$ and $\struct {X_2, \tau_2}$ be topological spaces.
Let $f: X_1 \to X_2$ be a mapping.
Let $\SS$ be an analytic sub-basis for $\tau_2$.
Suppose that:
:$\forall S \in \SS: f^{-1} \sqbrk S \in \tau_1$
where $f^{-1} \sqbrk S$ denotes the preimage of $S$ under $f$.
Then $f$ is continuous. | Define:
:$\ds \BB = \set {\bigcap \AA: \AA \subseteq \SS, \AA \text{ is finite} } \subseteq \powerset {X_2}$
Let $B \in \BB$.
Then there exists a finite subset $\AA \subseteq \SS$ such that:
:$\ds B = \bigcap \AA$
Hence:
{{begin-eqn}}
{{eqn | l = f^{-1} \sqbrk B
| r = f^{-1} \sqbrk {\bigcap \AA}
}}
{{eqn | r = \b... | Let $\struct {X_1, \tau_1}$ and $\struct {X_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $f: X_1 \to X_2$ be a [[Definition:Mapping|mapping]].
Let $\SS$ be an [[Definition:Analytic Sub-Basis|analytic sub-basis]] for $\tau_2$.
Suppose that:
:$\forall S \in \SS: f^{-1} \sqbrk S \in \tau_1$
w... | Define:
:$\ds \BB = \set {\bigcap \AA: \AA \subseteq \SS, \AA \text{ is finite} } \subseteq \powerset {X_2}$
Let $B \in \BB$.
Then [[Definition:Existential Quantifier|there exists]] a [[Definition:Finite Set|finite]] [[Definition:Subset|subset]] $\AA \subseteq \SS$ such that:
:$\ds B = \bigcap \AA$
Hence:
{{begin-eq... | Continuity Test using Sub-Basis/Proof 1 | https://proofwiki.org/wiki/Continuity_Test_using_Sub-Basis | https://proofwiki.org/wiki/Continuity_Test_using_Sub-Basis/Proof_1 | [
"Topological Bases",
"Continuous Mappings",
"Continuity Test using Sub-Basis"
] | [
"Definition:Topological Space",
"Definition:Mapping",
"Definition:Sub-Basis/Analytic Sub-Basis",
"Definition:Preimage/Mapping/Subset",
"Definition:Continuous Mapping (Topology)/Everywhere"
] | [
"Definition:Existential Quantifier",
"Definition:Finite Set",
"Definition:Subset",
"Preimage of Intersection under Mapping/General Result",
"General Intersection Property of Topological Space",
"Definition:Sub-Basis/Analytic Sub-Basis",
"Preimage of Union under Mapping/General Result",
"Definition:Con... |
proofwiki-6226 | Continuity Test using Sub-Basis | Let $\struct {X_1, \tau_1}$ and $\struct {X_2, \tau_2}$ be topological spaces.
Let $f: X_1 \to X_2$ be a mapping.
Let $\SS$ be an analytic sub-basis for $\tau_2$.
Suppose that:
:$\forall S \in \SS: f^{-1} \sqbrk S \in \tau_1$
where $f^{-1} \sqbrk S$ denotes the preimage of $S$ under $f$.
Then $f$ is continuous. | Let $\tau$ be the final topology on $X_2$ with respect to $f$.
By hypothesis, $\SS \subseteq \tau$.
By Synthetic Sub-Basis and Analytic Sub-Basis are Compatible, we have that $\tau_2$ is the topology generated by the synthetic sub-basis $\SS$.
By the definition of the generated topology, we have $\tau_2 \subseteq \tau$... | Let $\struct {X_1, \tau_1}$ and $\struct {X_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $f: X_1 \to X_2$ be a [[Definition:Mapping|mapping]].
Let $\SS$ be an [[Definition:Analytic Sub-Basis|analytic sub-basis]] for $\tau_2$.
Suppose that:
:$\forall S \in \SS: f^{-1} \sqbrk S \in \tau_1$
w... | Let $\tau$ be the [[Definition:Final Topology|final topology on $X_2$ with respect to $f$]].
[[Definition:By Hypothesis|By hypothesis]], $\SS \subseteq \tau$.
By [[Synthetic Sub-Basis and Analytic Sub-Basis are Compatible]], we have that $\tau_2$ is the [[Definition:Topology Generated by Synthetic Sub-Basis|topology ... | Continuity Test using Sub-Basis/Proof 2 | https://proofwiki.org/wiki/Continuity_Test_using_Sub-Basis | https://proofwiki.org/wiki/Continuity_Test_using_Sub-Basis/Proof_2 | [
"Topological Bases",
"Continuous Mappings",
"Continuity Test using Sub-Basis"
] | [
"Definition:Topological Space",
"Definition:Mapping",
"Definition:Sub-Basis/Analytic Sub-Basis",
"Definition:Preimage/Mapping/Subset",
"Definition:Continuous Mapping (Topology)/Everywhere"
] | [
"Definition:Final Topology",
"Definition:By Hypothesis",
"Synthetic Sub-Basis and Analytic Sub-Basis are Compatible",
"Definition:Topology Generated by Synthetic Sub-Basis",
"Definition:Topology Generated by Synthetic Sub-Basis/Definition 2",
"Definition:Final Topology",
"Definition:Continuous Mapping (... |
proofwiki-6227 | Product Formula for Sine | :$\ds \map \sin {n z} = 2^{n - 1} \prod_{k \mathop = 0}^{n - 1} \map \sin {z + \frac {k \pi} n}$ | From Gauss Multiplication Formula, we have:
:$\ds \forall z \notin \set {-\frac m n: m \in \N}: \prod_{k \mathop = 0}^{n - 1} \map \Gamma {z + \frac k n} = \paren {2 \pi}^{\paren {n - 1} / 2} n^{1/2 - n z} \map \Gamma {n z}$
Therefore:
:$\ds \map \Gamma {n z} = \paren {2 \pi}^{\paren {1 - n} / 2} n^{n z - 1/2} \prod_{k... | :$\ds \map \sin {n z} = 2^{n - 1} \prod_{k \mathop = 0}^{n - 1} \map \sin {z + \frac {k \pi} n}$ | From [[Gauss Multiplication Formula]], we have:
:$\ds \forall z \notin \set {-\frac m n: m \in \N}: \prod_{k \mathop = 0}^{n - 1} \map \Gamma {z + \frac k n} = \paren {2 \pi}^{\paren {n - 1} / 2} n^{1/2 - n z} \map \Gamma {n z}$
Therefore:
:$\ds \map \Gamma {n z} = \paren {2 \pi}^{\paren {1 - n} / 2} n^{n z - 1/2} \... | Product Formula for Sine/Proof 2 | https://proofwiki.org/wiki/Product_Formula_for_Sine | https://proofwiki.org/wiki/Product_Formula_for_Sine/Proof_2 | [
"Product Formula for Sine",
"Sine Function",
"Complex Roots of Unity",
"Named Theorems"
] | [] | [
"Gauss Multiplication Formula",
"Euler's Reflection Formula",
"Euler's Reflection Formula",
"Gamma Difference Equation",
"Gauss Multiplication Formula",
"Definition:Fraction/Numerator",
"Exponent Combination Laws/Product of Powers",
"Definition:Fraction/Denominator",
"Exponent Combination Laws/Produ... |
proofwiki-6228 | Final Topology is Topology | Let $X$ be a set.
Let $I$ be an indexing set.
Let $\family {\struct {Y_i, \tau_i} }_{i \mathop \in I}$ be an $I$-indexed family of topological spaces.
Let $\family {f_i: Y_i \to X}_{i \mathop \in I}$ be an $I$-indexed family of mappings.
Let $\tau$ be the final topology on $X$ with respect to $\family {f_i}_{i \mathop ... | Define:
:$\forall i \in I: \vartheta_i = \set {U \subseteq X: \map {f_i^{-1} } U \in \tau_i} \subseteq \powerset X$
Then, by the definition of intersection:
:$\ds \tau = \bigcap_{i \mathop \in I} \vartheta_i$
From the Intersection of Topologies is Topology, it suffices to show, for all $i \in I$, that $\vartheta_i$ is ... | Let $X$ be a [[Definition:Set|set]].
Let $I$ be an [[Definition:Indexing Set|indexing set]].
Let $\family {\struct {Y_i, \tau_i} }_{i \mathop \in I}$ be an [[Definition:Indexed Family|$I$-indexed family]] of [[Definition:Topological Space|topological spaces]].
Let $\family {f_i: Y_i \to X}_{i \mathop \in I}$ be an ... | Define:
:$\forall i \in I: \vartheta_i = \set {U \subseteq X: \map {f_i^{-1} } U \in \tau_i} \subseteq \powerset X$
Then, by the definition of [[Definition:Intersection of Family|intersection]]:
:$\ds \tau = \bigcap_{i \mathop \in I} \vartheta_i$
From the [[Intersection of Topologies is Topology]], it suffices to sho... | Final Topology is Topology | https://proofwiki.org/wiki/Final_Topology_is_Topology | https://proofwiki.org/wiki/Final_Topology_is_Topology | [
"Final Topology"
] | [
"Definition:Set",
"Definition:Indexing Set",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Mapping",
"Definition:Final Topology",
"Definition:Topology"
] | [
"Definition:Set Intersection/Family of Sets",
"Intersection of Topologies is Topology",
"Definition:Universal Quantifier",
"Definition:Topology",
"Axiom:Open Set Axioms",
"Definition:Topology",
"Definition:Topology",
"Definition:Topology",
"Definition:Topology",
"Axiom:Open Set Axioms"
] |
proofwiki-6229 | Subspace Topology is Initial Topology with respect to Inclusion Mapping | Let $\struct {X, \tau}$ be a topological space.
Let $Y$ be a non-empty subset of $X$.
Let $\iota: Y \to X$ be the inclusion mapping.
Let $\tau_Y$ be the initial topology on $Y$ with respect to $\iota$.
Then $\struct {Y, \tau_Y}$ is a topological subspace of $\struct {X, \tau}$.
That is:
:$\tau_Y = \set {U \cap Y: U \in... | By Initial Topology with respect to Mapping equals Set of Preimages, it follows that:
:$\tau_Y = \set {\iota^{-1} \sqbrk U: U \in \tau}$
From Preimage of Subset under Inclusion Mapping, we have:
:$\forall S \subseteq X: \iota^{-1} \sqbrk S = S \cap Y$
Hence the result.
{{qed}}
Category:Topological Subspaces
Category:In... | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $Y$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $X$.
Let $\iota: Y \to X$ be the [[Definition:Inclusion Mapping|inclusion mapping]].
Let $\tau_Y$ be the [[Definition:Initial Topology|initial topology]... | By [[Initial Topology with respect to Mapping equals Set of Preimages]], it follows that:
:$\tau_Y = \set {\iota^{-1} \sqbrk U: U \in \tau}$
From [[Preimage of Subset under Inclusion Mapping]], we have:
:$\forall S \subseteq X: \iota^{-1} \sqbrk S = S \cap Y$
Hence the result.
{{qed}}
[[Category:Topological Subspace... | Subspace Topology is Initial Topology with respect to Inclusion Mapping | https://proofwiki.org/wiki/Subspace_Topology_is_Initial_Topology_with_respect_to_Inclusion_Mapping | https://proofwiki.org/wiki/Subspace_Topology_is_Initial_Topology_with_respect_to_Inclusion_Mapping | [
"Topological Subspaces",
"Inclusion Mappings",
"Initial Topology"
] | [
"Definition:Topological Space",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Inclusion Mapping",
"Definition:Initial Topology",
"Definition:Topological Subspace"
] | [
"Initial Topology with respect to Mapping equals Set of Preimages",
"Preimage of Subset under Inclusion Mapping",
"Category:Topological Subspaces",
"Category:Inclusion Mappings",
"Category:Initial Topology"
] |
proofwiki-6230 | Topological Subspace is Topological Space | Let $\struct {X, \tau}$ be a topological space.
Let $H \subseteq X$ be a non-empty subset of $X$.
Let $\tau_H = \set {U \cap H: U \in \tau}$ be the subspace topology on $H$.
Then the topological subspace $\struct {H, \tau_H}$ is a topological space. | We verify the open set axioms for $\tau_H$ to be a topology on $H$.
=== {{Open-set-axiom|1|nolink}} ===
Let $\AA \subseteq \tau_H$.
It is to be shown that:
:$\ds \bigcup \AA \in \tau_H$
Define:
:$\ds \AA' = \set {V \in \tau: V \cap H \subseteq \bigcup \AA} \subseteq \tau$
Let:
:$\ds U = \bigcup \AA'$
By the definition ... | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq X$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $X$.
Let $\tau_H = \set {U \cap H: U \in \tau}$ be the [[Definition:Topological Subspace|subspace topology]] on $H$.
Then the [[Definition:... | We verify the [[Axiom:Open Set Axioms|open set axioms]] for $\tau_H$ to be a [[Definition:Topology|topology]] on $H$.
=== {{Open-set-axiom|1|nolink}} ===
Let $\AA \subseteq \tau_H$.
It is to be shown that:
:$\ds \bigcup \AA \in \tau_H$
Define:
:$\ds \AA' = \set {V \in \tau: V \cap H \subseteq \bigcup \AA} \subset... | Topological Subspace is Topological Space/Proof 1 | https://proofwiki.org/wiki/Topological_Subspace_is_Topological_Space | https://proofwiki.org/wiki/Topological_Subspace_is_Topological_Space/Proof_1 | [
"Topological Subspace is Topological Space",
"Topological Subspaces"
] | [
"Definition:Topological Space",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Definition:Topological Space"
] | [
"Axiom:Open Set Axioms",
"Definition:Topology",
"Definition:Topology",
"Intersection Distributes over Union",
"Union is Smallest Superset/Family of Sets",
"Set is Subset of Union/General Result",
"Set is Subset of Union/General Result",
"Set Intersection Preserves Subsets/Corollary",
"Union is Small... |
proofwiki-6231 | Topological Subspace is Topological Space | Let $\struct {X, \tau}$ be a topological space.
Let $H \subseteq X$ be a non-empty subset of $X$.
Let $\tau_H = \set {U \cap H: U \in \tau}$ be the subspace topology on $H$.
Then the topological subspace $\struct {H, \tau_H}$ is a topological space. | Follows directly from Subspace Topology is Initial Topology with respect to Inclusion Mapping.
{{qed}} | Let $\struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq X$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $X$.
Let $\tau_H = \set {U \cap H: U \in \tau}$ be the [[Definition:Topological Subspace|subspace topology]] on $H$.
Then the [[Definition:... | Follows directly from [[Subspace Topology is Initial Topology with respect to Inclusion Mapping]].
{{qed}} | Topological Subspace is Topological Space/Proof 2 | https://proofwiki.org/wiki/Topological_Subspace_is_Topological_Space | https://proofwiki.org/wiki/Topological_Subspace_is_Topological_Space/Proof_2 | [
"Topological Subspace is Topological Space",
"Topological Subspaces"
] | [
"Definition:Topological Space",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Topological Subspace",
"Definition:Topological Subspace",
"Definition:Topological Space"
] | [
"Subspace Topology is Initial Topology with respect to Inclusion Mapping"
] |
proofwiki-6232 | Initial Topology with respect to Mapping equals Set of Preimages | Let $X$ be a set.
Let $\struct {Y, \tau_Y}$ be a topological space.
Let $f: X \to Y$ be a mapping.
Let $\tau_X$ be the initial topology on $X$ {{WRT}} $f$.
Then:
:$\tau_X = \set {f^{-1} \sqbrk U: U \in \tau_Y}$ | Define:
:$\tau = \set {f^{-1} \sqbrk U: U \in \tau_Y}$
By definition, $\tau_X$ is the topology generated by $\tau$.
Therefore:
:$\tau \subseteq \tau_X$
If $\tau$ is a topology on $X$, then it follows from the definition of the generated topology that:
:$\tau_X \subseteq \tau$
By definition of set equality:
:$\tau_X = \... | Let $X$ be a [[Definition:Set|set]].
Let $\struct {Y, \tau_Y}$ be a [[Definition:Topological Space|topological space]].
Let $f: X \to Y$ be a [[Definition:Mapping|mapping]].
Let $\tau_X$ be the [[Definition:Initial Topology|initial topology]] on $X$ {{WRT}} $f$.
Then:
:$\tau_X = \set {f^{-1} \sqbrk U: U \in \tau_Y... | Define:
:$\tau = \set {f^{-1} \sqbrk U: U \in \tau_Y}$
By definition, $\tau_X$ is the [[Definition:Topology Generated by Synthetic Sub-Basis|topology generated]] by $\tau$.
Therefore:
:$\tau \subseteq \tau_X$
If $\tau$ is a [[Definition:Topology|topology]] on $X$, then it follows from the definition of the [[Defini... | Initial Topology with respect to Mapping equals Set of Preimages | https://proofwiki.org/wiki/Initial_Topology_with_respect_to_Mapping_equals_Set_of_Preimages | https://proofwiki.org/wiki/Initial_Topology_with_respect_to_Mapping_equals_Set_of_Preimages | [
"Topology"
] | [
"Definition:Set",
"Definition:Topological Space",
"Definition:Mapping",
"Definition:Initial Topology"
] | [
"Definition:Topology Generated by Synthetic Sub-Basis",
"Definition:Topology",
"Definition:Topology Generated by Synthetic Sub-Basis/Definition 2",
"Definition:Set Equality/Definition 2",
"Definition:Topology",
"Axiom:Open Set Axioms",
"Definition:Topology",
"Definition:Topology",
"Definition:Set Eq... |
proofwiki-6233 | Sommerfeld-Watson Transform | Let $\map f z$ be a mapping with isolated poles.
{{explain|Explain the context in which this theorem is placed. For example, what is the domain and range of $f$? One supposes $\C$ but it needs to be made clear.}}
Let $f$ go to zero faster than $\dfrac 1 {\size z}$ as $\size z \to \infty$.
{{explain|"faster"}}
Let $C$ b... | From Cauchy's Residue Theorem:
{{begin-eqn}}
{{eqn | l = \oint_C \map f z \rd z
| r = 2 \pi i \, \sum \limits_{z_k} R_k(z_k)
| c =
}}
{{eqn | r = 2 \pi i \, \sum_{z_k} \lim_{z \mathop \to z_k} \paren {\paren {z - z_k} \frac {\map f z} {\sin \pi z} }
| c =
}}
{{end-eqn}}
This is for poles $z_k$ at or... | Let $\map f z$ be a [[Definition:Mapping|mapping]] with isolated poles.
{{explain|Explain the context in which this theorem is placed. For example, what is the domain and range of $f$? One supposes $\C$ but it needs to be made clear.}}
Let $f$ go to zero faster than $\dfrac 1 {\size z}$ as $\size z \to \infty$.
{{ex... | From [[Cauchy's Residue Theorem]]:
{{begin-eqn}}
{{eqn | l = \oint_C \map f z \rd z
| r = 2 \pi i \, \sum \limits_{z_k} R_k(z_k)
| c =
}}
{{eqn | r = 2 \pi i \, \sum_{z_k} \lim_{z \mathop \to z_k} \paren {\paren {z - z_k} \frac {\map f z} {\sin \pi z} }
| c =
}}
{{end-eqn}}
This is for poles $z_k... | Sommerfeld-Watson Transform | https://proofwiki.org/wiki/Sommerfeld-Watson_Transform | https://proofwiki.org/wiki/Sommerfeld-Watson_Transform | [
"Complex Analysis"
] | [
"Definition:Mapping"
] | [
"Cauchy's Residue Theorem",
"Category:Complex Analysis"
] |
proofwiki-6234 | Linear Transformation of Arithmetic Mean | Let $D = \set {x_0, x_1, x_2, \ldots, x_n}$ be a set of real data describing a quantitative variable.
Let $\overline x$ be the arithmetic mean of the data in $D$.
Let $T: \R \to \R$ be a linear transformation such that:
:$\forall i \in \set {0, 1, \ldots, n}: \map T {x_i} = \lambda x_i + \gamma$
Let $T \sqbrk D$ be the... | Follows from the definition of arithmetic mean and from Summation is Linear.
{{qed}} | Let $D = \set {x_0, x_1, x_2, \ldots, x_n}$ be a [[Definition:Set|set]] of [[Definition:Real Number|real]] data describing a [[Definition:Quantitative Variable|quantitative variable]].
Let $\overline x$ be the [[Definition:Arithmetic Mean|arithmetic mean]] of the data in $D$.
Let $T: \R \to \R$ be a [[Definition:Line... | Follows from the [[Definition:Arithmetic Mean|definition of arithmetic mean]] and from [[Summation is Linear]].
{{qed}} | Linear Transformation of Arithmetic Mean/Proof 1 | https://proofwiki.org/wiki/Linear_Transformation_of_Arithmetic_Mean | https://proofwiki.org/wiki/Linear_Transformation_of_Arithmetic_Mean/Proof_1 | [
"Arithmetic Mean",
"Linear Transformations",
"Linear Transformation of Arithmetic Mean"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Variable/Descriptive Statistics/Quantitative Variable",
"Definition:Arithmetic Mean",
"Definition:Linear Transformation",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Arithmetic Mean"
] | [
"Definition:Arithmetic Mean",
"Summation is Linear"
] |
proofwiki-6235 | Linear Transformation of Arithmetic Mean | Let $D = \set {x_0, x_1, x_2, \ldots, x_n}$ be a set of real data describing a quantitative variable.
Let $\overline x$ be the arithmetic mean of the data in $D$.
Let $T: \R \to \R$ be a linear transformation such that:
:$\forall i \in \set {0, 1, \ldots, n}: \map T {x_i} = \lambda x_i + \gamma$
Let $T \sqbrk D$ be the... | This is a direct application of Expectation is Linear.
{{qed}} | Let $D = \set {x_0, x_1, x_2, \ldots, x_n}$ be a [[Definition:Set|set]] of [[Definition:Real Number|real]] data describing a [[Definition:Quantitative Variable|quantitative variable]].
Let $\overline x$ be the [[Definition:Arithmetic Mean|arithmetic mean]] of the data in $D$.
Let $T: \R \to \R$ be a [[Definition:Line... | This is a direct application of [[Expectation is Linear]].
{{qed}} | Linear Transformation of Arithmetic Mean/Proof 2 | https://proofwiki.org/wiki/Linear_Transformation_of_Arithmetic_Mean | https://proofwiki.org/wiki/Linear_Transformation_of_Arithmetic_Mean/Proof_2 | [
"Arithmetic Mean",
"Linear Transformations",
"Linear Transformation of Arithmetic Mean"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Variable/Descriptive Statistics/Quantitative Variable",
"Definition:Arithmetic Mean",
"Definition:Linear Transformation",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Arithmetic Mean"
] | [
"Expectation is Linear"
] |
proofwiki-6236 | Category of Pointed Sets is Category | Let $\mathbf{Set}_*$ be the category of pointed sets.
Then $\mathbf{Set}_*$ is a metacategory. | The axioms $(\text C 1)$ to $(\text C 3)$ are checked for a metacategory.
Pick any two morphisms $f : \struct {A, a} \to \struct {B, b}$ and $g : \struct {B, b} \to \struct {C, c}$ from $\mathbf{Set}_*$.
By the definition of composition in the category of pointed sets:
:$\map {\paren {g \circ f} } a = \map g {\map f a}... | Let $\mathbf{Set}_*$ be the [[Definition:Category of Pointed Sets|category of pointed sets]].
Then $\mathbf{Set}_*$ is a [[Definition:Metacategory|metacategory]]. | The axioms $(\text C 1)$ to $(\text C 3)$ are checked for a [[Definition:Metacategory|metacategory]].
Pick any two [[Definition:Morphism|morphisms]] $f : \struct {A, a} \to \struct {B, b}$ and $g : \struct {B, b} \to \struct {C, c}$ from $\mathbf{Set}_*$.
By the definition of [[Definition:Composition of Morphisms|co... | Category of Pointed Sets is Category | https://proofwiki.org/wiki/Category_of_Pointed_Sets_is_Category | https://proofwiki.org/wiki/Category_of_Pointed_Sets_is_Category | [
"Category Theory"
] | [
"Definition:Category of Pointed Sets",
"Definition:Metacategory"
] | [
"Definition:Metacategory",
"Definition:Morphism",
"Definition:Composition of Morphisms",
"Definition:Category of Pointed Sets",
"Definition:Pointed Mapping",
"Definition:Pointed Mapping",
"Definition:Associative Operation",
"Composition of Mappings is Associative",
"Definition:Object (Category Theor... |
proofwiki-6237 | Closed Set in Topological Subspace/Corollary | Let $H$ be closed in $T$.
Then $V \subseteq H$ is closed in $T'$ {{iff}} $V$ is closed in $T$. | Let $V \subseteq H$ be closed in $T'$.
Then, from Closed Set in Topological Subspace, $V = H \cap V$ is closed in $T'$.
If $V$ is closed in $T'$ then $V = H \cap W$ where $W$ is closed in $T$.
Since $H$ is closed in $T$, it follows by Topology Defined by Closed Sets that $V$ is closed in $T$.
{{qed}} | Let $H$ be [[Definition:Closed Set (Topology)|closed]] in $T$.
Then $V \subseteq H$ is [[Definition:Closed Set (Topology)|closed]] in $T'$ {{iff}} $V$ is [[Definition:Closed Set (Topology)|closed]] in $T$. | Let $V \subseteq H$ be [[Definition:Closed Set (Topology)|closed]] in $T'$.
Then, from [[Closed Set in Topological Subspace]], $V = H \cap V$ is [[Definition:Closed Set (Topology)|closed]] in $T'$.
If $V$ is [[Definition:Closed Set (Topology)|closed]] in $T'$ then $V = H \cap W$ where $W$ is [[Definition:Closed Set (... | Closed Set in Topological Subspace/Corollary | https://proofwiki.org/wiki/Closed_Set_in_Topological_Subspace/Corollary | https://proofwiki.org/wiki/Closed_Set_in_Topological_Subspace/Corollary | [
"Closed Sets",
"Topological Subspaces"
] | [
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology"
] | [
"Definition:Closed Set/Topology",
"Closed Set in Topological Subspace",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Topology Defined by Closed Sets",
"Definition:Closed Set/Topology"
] |
proofwiki-6238 | Unit Interval is Path-Connected in Real Numbers | Let $\R$ be the real number line with the usual (Euclidean} metric.
The closed unit interval $\I = \closedint 0 1$ is a path-connected metric subspace of $\R$. | Follows directly from Subset of Real Numbers is Path-Connected iff Interval.
{{qed}} | Let $\R$ be the [[Definition:Real Number Line with Euclidean Metric|real number line with the usual (Euclidean} metric]].
The [[Definition:Closed Unit Interval|closed unit interval]] $\I = \closedint 0 1$ is a [[Definition:Path-Connected Metric Subspace|path-connected metric subspace]] of $\R$. | Follows directly from [[Subset of Real Numbers is Path-Connected iff Interval]].
{{qed}} | Unit Interval is Path-Connected in Real Numbers | https://proofwiki.org/wiki/Unit_Interval_is_Path-Connected_in_Real_Numbers | https://proofwiki.org/wiki/Unit_Interval_is_Path-Connected_in_Real_Numbers | [
"Path-Connected Sets",
"Real Intervals"
] | [
"Definition:Euclidean Metric/Real Number Line",
"Definition:Real Interval/Unit Interval/Closed",
"Definition:Path-Connected/Metric Space/Subset"
] | [
"Subset of Real Numbers is Path-Connected iff Interval"
] |
proofwiki-6239 | Subset of Real Numbers is Path-Connected iff Interval | Let $\R$ be the real number line considered as an Euclidean space.
Let $S \subseteq \R$ be a subset of $\R$.
Then $S$ is a path-connected metric subspace of $\R$ {{iff}} $S$ is a real interval. | === Necessary Condition ===
Let $S$ be a path-connected metric subspace of $\R$.
From Path-Connected Space is Connected, it follows that $S$ is connected.
From Subset of Real Numbers is Interval iff Connected, it follows that $S$ is a real interval.
{{qed|lemma}} | Let $\R$ be the [[Definition:Real Number Line|real number line]] considered as an [[Definition:Euclidean Space|Euclidean space]].
Let $S \subseteq \R$ be a [[Definition:Subset|subset]] of $\R$.
Then $S$ is a [[Definition:Path-Connected Metric Subspace|path-connected metric subspace]] of $\R$ {{iff}} $S$ is a [[Defin... | === Necessary Condition ===
Let $S$ be a [[Definition:Path-Connected Metric Subspace|path-connected metric subspace]] of $\R$.
From [[Path-Connected Space is Connected]], it follows that $S$ is [[Definition:Connected Set (Topology)|connected]].
From [[Subset of Real Numbers is Interval iff Connected]], it follows th... | Subset of Real Numbers is Path-Connected iff Interval | https://proofwiki.org/wiki/Subset_of_Real_Numbers_is_Path-Connected_iff_Interval | https://proofwiki.org/wiki/Subset_of_Real_Numbers_is_Path-Connected_iff_Interval | [
"Path-Connected Spaces"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space",
"Definition:Subset",
"Definition:Path-Connected/Metric Space/Subset",
"Definition:Real Interval"
] | [
"Definition:Path-Connected/Metric Space/Subset",
"Path-Connected Space is Connected",
"Definition:Connected Set (Topology)",
"Subset of Real Numbers is Interval iff Connected",
"Definition:Real Interval",
"Definition:Real Interval"
] |
proofwiki-6240 | Size of Linearly Independent Subset is at Most Size of Finite Generator | Let $R$ be a division ring.
Let $V$ be an $R$-vector space.
Let $F \subseteq V$ be a finite generator of $V$ over $R$.
Let $L \subseteq V$ be linearly independent over $R$.
Then:
:$\size L \le \size F$ | We first consider the case where $L$ is finite.
Let $S \subseteq \N$ be the set of all $n \in \N$ such that:
:For every finite generator $F$ of $V$, if $\card {L \setminus F} \le n$, then $\card L \le \card F$
where:
:$L \setminus F$ denotes the set difference between $L$ and $F$
:$\card L$ and $\card F$ denote the car... | Let $R$ be a [[Definition:Division Ring|division ring]].
Let $V$ be an [[Definition:Vector Space|$R$-vector space]].
Let $F \subseteq V$ be a [[Definition:Finite Set|finite]] [[Definition:Generator of Module|generator]] of $V$ over $R$.
Let $L \subseteq V$ be [[Definition:Linearly Independent Set|linearly independen... | We first consider the case where $L$ is [[Definition:Finite Set|finite]].
Let $S \subseteq \N$ be the set of all $n \in \N$ such that:
:For every [[Definition:Finite Set|finite]] [[Definition:Generator of Vector Space|generator]] $F$ of $V$, if $\card {L \setminus F} \le n$, then $\card L \le \card F$
where:
:$L \setm... | Size of Linearly Independent Subset is at Most Size of Finite Generator/Proof 1 | https://proofwiki.org/wiki/Size_of_Linearly_Independent_Subset_is_at_Most_Size_of_Finite_Generator | https://proofwiki.org/wiki/Size_of_Linearly_Independent_Subset_is_at_Most_Size_of_Finite_Generator/Proof_1 | [
"Vector Spaces",
"Size of Linearly Independent Subset is at Most Size of Finite Generator"
] | [
"Definition:Division Ring",
"Definition:Vector Space",
"Definition:Finite Set",
"Definition:Generator of Module",
"Definition:Linearly Independent/Set"
] | [
"Definition:Finite Set",
"Definition:Finite Set",
"Definition:Generator of Vector Space",
"Definition:Set Difference",
"Definition:Cardinality",
"Principle of Finite Induction",
"Cardinality of Empty Set",
"Set Difference with Superset is Empty Set",
"Cardinality of Subset of Finite Set",
"Definit... |
proofwiki-6241 | Size of Linearly Independent Subset is at Most Size of Finite Generator | Let $R$ be a division ring.
Let $V$ be an $R$-vector space.
Let $F \subseteq V$ be a finite generator of $V$ over $R$.
Let $L \subseteq V$ be linearly independent over $R$.
Then:
:$\size L \le \size F$ | Let $S \subseteq \N$ be the set of all natural numbers $n \in \N$ such that:
:For any finite generator $F$ of $V$ over $R$, if $\card {F \cap L} \ge n$, then $\card L \le \card F$.
It is to be demonstrated that $S = \N$.
That is, that $\card {F \cap L} \ge n \implies \card L \le \card F$ for all $n \in \N$.
By Intersec... | Let $R$ be a [[Definition:Division Ring|division ring]].
Let $V$ be an [[Definition:Vector Space|$R$-vector space]].
Let $F \subseteq V$ be a [[Definition:Finite Set|finite]] [[Definition:Generator of Module|generator]] of $V$ over $R$.
Let $L \subseteq V$ be [[Definition:Linearly Independent Set|linearly independen... | Let $S \subseteq \N$ be the set of all [[Definition:Natural Numbers|natural numbers]] $n \in \N$ such that:
:For any [[Definition:Finite Set|finite]] [[Definition:Generator of Module|generator]] $F$ of $V$ over $R$, if $\card {F \cap L} \ge n$, then $\card L \le \card F$.
It is to be demonstrated that $S = \N$.
That ... | Size of Linearly Independent Subset is at Most Size of Finite Generator/Proof 2 | https://proofwiki.org/wiki/Size_of_Linearly_Independent_Subset_is_at_Most_Size_of_Finite_Generator | https://proofwiki.org/wiki/Size_of_Linearly_Independent_Subset_is_at_Most_Size_of_Finite_Generator/Proof_2 | [
"Vector Spaces",
"Size of Linearly Independent Subset is at Most Size of Finite Generator"
] | [
"Definition:Division Ring",
"Definition:Vector Space",
"Definition:Finite Set",
"Definition:Generator of Module",
"Definition:Linearly Independent/Set"
] | [
"Definition:Natural Numbers",
"Definition:Finite Set",
"Definition:Generator of Module",
"Intersection is Subset",
"Cardinality of Subset of Finite Set",
"Definition:Vacuous Truth",
"Definition:Non-Empty Set",
"Well-Ordering Principle",
"Definition:Smallest Element",
"Cardinality of Subset of Fini... |
proofwiki-6242 | Size of Linearly Independent Subset is at Most Size of Finite Generator | Let $R$ be a division ring.
Let $V$ be an $R$-vector space.
Let $F \subseteq V$ be a finite generator of $V$ over $R$.
Let $L \subseteq V$ be linearly independent over $R$.
Then:
:$\size L \le \size F$ | Let $\alpha_1, \alpha_2, \ldots, \alpha_n$ be a generator of $V$.
Let $\xi_1, \xi_2, \ldots, \xi_r$ be a linearly independent set of elements of $V$.
Hence the sequence $\sequence {\xi_1, \alpha_1, \alpha_2, \ldots, \alpha_n}$ is a linearly dependent sequence of elements of $V$.
One of these elements, which cannot be $... | Let $R$ be a [[Definition:Division Ring|division ring]].
Let $V$ be an [[Definition:Vector Space|$R$-vector space]].
Let $F \subseteq V$ be a [[Definition:Finite Set|finite]] [[Definition:Generator of Module|generator]] of $V$ over $R$.
Let $L \subseteq V$ be [[Definition:Linearly Independent Set|linearly independen... | Let $\alpha_1, \alpha_2, \ldots, \alpha_n$ be a [[Definition:Generator of Vector Space|generator]] of $V$.
Let $\xi_1, \xi_2, \ldots, \xi_r$ be a [[Definition:Linearly Independent Set|linearly independent set]] of [[Definition:Element|elements]] of $V$.
Hence the [[Definition:Sequence|sequence]] $\sequence {\xi_1, \a... | Size of Linearly Independent Subset is at Most Size of Finite Generator/Proof 3 | https://proofwiki.org/wiki/Size_of_Linearly_Independent_Subset_is_at_Most_Size_of_Finite_Generator | https://proofwiki.org/wiki/Size_of_Linearly_Independent_Subset_is_at_Most_Size_of_Finite_Generator/Proof_3 | [
"Vector Spaces",
"Size of Linearly Independent Subset is at Most Size of Finite Generator"
] | [
"Definition:Division Ring",
"Definition:Vector Space",
"Definition:Finite Set",
"Definition:Generator of Module",
"Definition:Linearly Independent/Set"
] | [
"Definition:Generator of Vector Space",
"Definition:Linearly Independent/Set",
"Definition:Element",
"Definition:Sequence",
"Definition:Linearly Dependent/Sequence",
"Definition:Element",
"Definition:Element",
"Definition:Linear Combination/Subset",
"Definition:Element",
"Definition:Element",
"D... |
proofwiki-6243 | Intermediate Value Theorem/Corollary | Let $0 \in \R$ lie between $\map f a$ and $\map f b$.
That is, either:
:$\map f a < 0 < \map f b$
or:
:$\map f b < 0 < \map f a$
Then $f$ has a root in $\openint a b$. | Follows directly from the Intermediate Value Theorem and from the definition of root.
{{qed}} | Let $0 \in \R$ lie between $\map f a$ and $\map f b$.
That is, either:
:$\map f a < 0 < \map f b$
or:
:$\map f b < 0 < \map f a$
Then $f$ has a [[Definition:Root of Function|root]] in $\openint a b$. | Follows directly from the [[Intermediate Value Theorem]] and from the definition of [[Definition:Root of Function|root]].
{{qed}} | Intermediate Value Theorem/Corollary | https://proofwiki.org/wiki/Intermediate_Value_Theorem/Corollary | https://proofwiki.org/wiki/Intermediate_Value_Theorem/Corollary | [
"Analysis"
] | [
"Definition:Root of Mapping"
] | [
"Intermediate Value Theorem",
"Definition:Root of Mapping"
] |
proofwiki-6244 | Real Number Line with Point Removed is Not Path-Connected | Let $\R$ be the real number line considered as an Euclidean space.
Let $x \in \R$ be a real number.
Then $\R \setminus \set x$, where $\setminus$ denotes set difference, is not path-connected. | We have that $x - 1$ and $x + 1$ are both real numbers, so:
:$x - 1 \in \R \setminus \set x$
:$x + 1 \in \R \setminus \set x$
Let $\I := \closedint 0 1$ be the closed unit interval.
{{AimForCont}} there exists a path $f: \I \to \R \setminus \set x$ from $x - 1$ to $x + 1$.
Then by Image of Real Interval under Continuou... | Let $\R$ be the [[Definition:Real Number Line with Euclidean Topology|real number line]] considered as an [[Definition:Euclidean Space|Euclidean space]].
Let $x \in \R$ be a [[Definition:Real Number|real number]].
Then $\R \setminus \set x$, where $\setminus$ denotes [[Definition:Set Difference|set difference]], is ... | We have that $x - 1$ and $x + 1$ are both [[Definition:Real Number|real numbers]], so:
:$x - 1 \in \R \setminus \set x$
:$x + 1 \in \R \setminus \set x$
Let $\I := \closedint 0 1$ be the [[Definition:Closed Unit Interval|closed unit interval]].
{{AimForCont}} there exists a [[Definition:Path (Topology)|path]] $f: \I ... | Real Number Line with Point Removed is Not Path-Connected | https://proofwiki.org/wiki/Real_Number_Line_with_Point_Removed_is_Not_Path-Connected | https://proofwiki.org/wiki/Real_Number_Line_with_Point_Removed_is_Not_Path-Connected | [
"Real Number Line with Euclidean Topology",
"Examples of Path-Connected Metric Spaces"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Euclidean Space",
"Definition:Real Number",
"Definition:Set Difference",
"Definition:Path-Connected/Metric Space/Subset"
] | [
"Definition:Real Number",
"Definition:Real Interval/Unit Interval/Closed",
"Definition:Path (Topology)",
"Image of Real Interval under Continuous Real Function is Real Interval",
"Definition:Set Difference",
"Definition:Contradiction",
"Definition:Path-Connected/Metric Space/Subset"
] |
proofwiki-6245 | Euclidean Space is Path-Connected | Let $\R^n$ be the $n$-dimensional Euclidean space for $n \in \N$ a natural number.
Then $\R^n$ is path-connected. | Let $\mathbf x, \mathbf y \in \R^n$ be arbitrary points of $\R^n$.
Define $l: \closedint 0 1 \to \R^n$ by:
:$\map l t = \paren {1 - t} \mathbf x + t \mathbf y$
Then $\map l 0 = 1 \mathbf x + 0 \mathbf y = \mathbf x$, whereas $\map l 1 = 0 \mathbf x + 1 \mathbf y = \mathbf y$.
Finally, it remains to show that $l$ is con... | Let $\R^n$ be the [[Definition:Euclidean Space|$n$-dimensional Euclidean space]] for $n \in \N$ a [[Definition:Natural Number|natural number]].
Then $\R^n$ is [[Definition:Path-Connected Metric Space|path-connected]]. | Let $\mathbf x, \mathbf y \in \R^n$ be arbitrary points of $\R^n$.
Define $l: \closedint 0 1 \to \R^n$ by:
:$\map l t = \paren {1 - t} \mathbf x + t \mathbf y$
Then $\map l 0 = 1 \mathbf x + 0 \mathbf y = \mathbf x$, whereas $\map l 1 = 0 \mathbf x + 1 \mathbf y = \mathbf y$.
Finally, it remains to show that $l$ i... | Euclidean Space is Path-Connected | https://proofwiki.org/wiki/Euclidean_Space_is_Path-Connected | https://proofwiki.org/wiki/Euclidean_Space_is_Path-Connected | [
"Euclidean Spaces",
"Examples of Path-Connected Metric Spaces"
] | [
"Definition:Euclidean Space",
"Definition:Natural Numbers",
"Definition:Path-Connected/Metric Space"
] | [
"Definition:Continuous Mapping (Metric Space)",
"Definition:Euclidean Norm",
"Triangle Inequality/Vectors in Euclidean Space",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Path (Topology)",
"Definition:Path-Connected/Metric Space"
] |
proofwiki-6246 | Coequalizer is Epimorphism | Let $\mathbf C$ be a metacategory.
Let $q: D \to Q$ be the coequalizer of two morphisms $f, g: C \to D$.
Then $q$ is an epimorphism. | Follows directly from Equalizer is Monomorphism and the Duality Principle.
{{qed}} | Let $\mathbf C$ be a [[Definition:Metacategory|metacategory]].
Let $q: D \to Q$ be the [[Definition:Coequalizer|coequalizer]] of two [[Definition:Morphism|morphisms]] $f, g: C \to D$.
Then $q$ is an [[Definition:Epimorphism (Category Theory)|epimorphism]]. | Follows directly from [[Equalizer is Monomorphism]] and the [[Duality Principle (Category Theory)|Duality Principle]].
{{qed}} | Coequalizer is Epimorphism | https://proofwiki.org/wiki/Coequalizer_is_Epimorphism | https://proofwiki.org/wiki/Coequalizer_is_Epimorphism | [
"Epimorphisms"
] | [
"Definition:Metacategory",
"Definition:Coequalizer",
"Definition:Morphism",
"Definition:Epimorphism (Category Theory)"
] | [
"Equalizer is Monomorphism",
"Duality Principle (Category Theory)"
] |
proofwiki-6247 | Ordering on Multiindices is Partial Order | Let $Z$ be the set of multiindices indexed by a set $J$.
The ordering on $Z$ is a partial ordering. | Let $\le$ denote the ordering on integers.
Let $\preceq$ denote the ordeing on multiindices.
Recall that if $k = \family {k_j}_{j \mathop \in J}$ and $\ell = \family {\ell_j}_{j \mathop \in J}$ are multiindices, then $k \le \ell$ if $k_j \preceq \ell_j$ for all $j \in J$.
Let $k$ and $\ell$ be as above, and let $m = \f... | Let $Z$ be the [[Definition:Set|set]] of [[Definition:Multiindex|multiindices]] [[Definition:Indexing Set|indexed]] by a set $J$.
The [[Definition:Ordering on Multiindices|ordering on $Z$]] is a [[Definition:Partial Ordering|partial ordering]]. | Let $\le$ denote the [[Definition:Ordering on Integers|ordering on integers]].
Let $\preceq$ denote the ordeing on multiindices.
Recall that if $k = \family {k_j}_{j \mathop \in J}$ and $\ell = \family {\ell_j}_{j \mathop \in J}$ are multiindices, then $k \le \ell$ if $k_j \preceq \ell_j$ for all $j \in J$.
Let $k$ ... | Ordering on Multiindices is Partial Order | https://proofwiki.org/wiki/Ordering_on_Multiindices_is_Partial_Order | https://proofwiki.org/wiki/Ordering_on_Multiindices_is_Partial_Order | [
"Polynomial Theory"
] | [
"Definition:Set",
"Definition:Multiindex",
"Definition:Indexing Set",
"Definition:Ordering on Multiindices",
"Definition:Partial Ordering"
] | [
"Definition:Ordering on Integers",
"Integers under Addition form Totally Ordered Group",
"Definition:Integer",
"Definition:Totally Ordered Set",
"Definition:Ordering on Integers",
"Definition:Reflexive Relation",
"Definition:Antisymmetric Relation",
"Definition:Transitive Relation",
"Definition:Refl... |
proofwiki-6248 | Euclidean Space without Origin is Path-Connected | Let $n \in \Z: n \ge 2$.
Let $\R^n$ be the $n$-dimensional Euclidean space.
Let $\R^n \setminus \set {\mathbf 0}$ be $\R^n$ with the origin removed.
Then $\R^n \setminus \set {\mathbf 0}$ is path-connected. | {{proof wanted|Another day, this is tedious.}} | Let $n \in \Z: n \ge 2$.
Let $\R^n$ be the [[Definition:Euclidean Space|$n$-dimensional Euclidean space]].
Let $\R^n \setminus \set {\mathbf 0}$ be $\R^n$ with the [[Definition:Origin|origin]] removed.
Then $\R^n \setminus \set {\mathbf 0}$ is [[Definition:Path-Connected Metric Subspace|path-connected]]. | {{proof wanted|Another day, this is tedious.}} | Euclidean Space without Origin is Path-Connected | https://proofwiki.org/wiki/Euclidean_Space_without_Origin_is_Path-Connected | https://proofwiki.org/wiki/Euclidean_Space_without_Origin_is_Path-Connected | [
"Path-Connected Spaces",
"Metric Spaces"
] | [
"Definition:Euclidean Space",
"Definition:Coordinate System/Origin",
"Definition:Path-Connected/Metric Space/Subset"
] | [] |
proofwiki-6249 | Multiindices under Addition form Commutative Monoid | Let $Z$ be the set of multiindices.
Let $+$ denote the addition of multiindices.
Then $\left({Z, +}\right)$ is a commutative monoid. | We check each of the axioms in turn.
Let $k = \left \langle {k_j}\right \rangle_{j \in J}$, $\ell = \left \langle {\ell_j}\right \rangle_{j \in J}$ and $m = \left \langle {m_j}\right \rangle_{j \in J}$ be multiindices. | Let $Z$ be the [[Definition:Set|set]] of [[Definition:Multiindex|multiindices]].
Let $+$ denote the [[Definition:Addition/Multiindices|addition of multiindices]].
Then $\left({Z, +}\right)$ is a [[Definition:Commutative Monoid|commutative monoid]]. | We check each of the axioms in turn.
Let $k = \left \langle {k_j}\right \rangle_{j \in J}$, $\ell = \left \langle {\ell_j}\right \rangle_{j \in J}$ and $m = \left \langle {m_j}\right \rangle_{j \in J}$ be [[Definition:Multiindex|multiindices]]. | Multiindices under Addition form Commutative Monoid | https://proofwiki.org/wiki/Multiindices_under_Addition_form_Commutative_Monoid | https://proofwiki.org/wiki/Multiindices_under_Addition_form_Commutative_Monoid | [
"Polynomial Theory"
] | [
"Definition:Set",
"Definition:Multiindex",
"Definition:Addition/Multiindices",
"Definition:Commutative Monoid"
] | [
"Definition:Multiindex",
"Definition:Multiindex"
] |
proofwiki-6250 | Composite of Continuous Mappings is Continuous/Corollary | Let $T_1, T_2, T_3$ each be one of:
:metric spaces
:the complex plane
:the real number line
Let $f: T_1 \to T_2$ and $g: T_2 \to T_3$ be continuous mappings.
Then the composite mapping $g \circ f: T_1 \to T_3$ is continuous. | These follow directly from:
* Real Number Line is Metric Space
* Complex Plane is Metric Space
* Metric Induces Topology
{{explain|How the claim follows from these links? Noting is proved about the continuity of the composition.}}
{{qed}} | Let $T_1, T_2, T_3$ each be one of:
:[[Definition:Metric Space|metric spaces]]
:the [[Definition:Complex Plane|complex plane]]
:the [[Definition:Real Number Line|real number line]]
Let $f: T_1 \to T_2$ and $g: T_2 \to T_3$ be [[Definition:Continuous Mapping (Topology)|continuous mappings]].
Then the [[Definition:C... | These follow directly from:
* [[Real Number Line is Metric Space]]
* [[Complex Plane is Metric Space]]
* [[Metric Induces Topology]]
{{explain|How the claim follows from these links? Noting is proved about the continuity of the composition.}}
{{qed}} | Composite of Continuous Mappings is Continuous/Corollary | https://proofwiki.org/wiki/Composite_of_Continuous_Mappings_is_Continuous/Corollary | https://proofwiki.org/wiki/Composite_of_Continuous_Mappings_is_Continuous/Corollary | [
"Composite of Continuous Mappings is Continuous",
"Continuous Real Functions",
"Continuous Complex Functions",
"Continuous Mappings on Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Complex Number/Complex Plane",
"Definition:Real Number/Real Number Line",
"Definition:Continuous Mapping (Topology)",
"Definition:Composition of Mappings",
"Definition:Continuous Mapping (Topology)"
] | [
"Real Number Line is Metric Space",
"Complex Plane is Metric Space",
"Metric Induces Topology"
] |
proofwiki-6251 | Binomial Theorem/Multiindex | Let $\alpha$ be a multiindex, indexed by $\set {1, \ldots, n}$ such that $\alpha_j \ge 0$ for $j = 1, \ldots, n$.
Let $x = \tuple {x_1, \ldots, x_n}$ and $y = \tuple {y_1, \ldots, y_n}$ be ordered tuples of real numbers.
Then:
:$\ds \paren {x + y}^\alpha = \sum_{0 \mathop \le \beta \mathop \le \alpha} \dbinom \alpha \b... | First of all, by definition of multiindexed powers:
:$\ds \paren {x + y}^\alpha = \prod_{k \mathop = 1}^n \paren {x_k + y_k}^{\alpha_k}$
Then:
{{begin-eqn}}
{{eqn | l = \paren {x + y}^\alpha
| r = \prod_{k \mathop = 1}^n \sum_{\beta_k \mathop = 0}^{\alpha_k} \dbinom {\alpha_k} {\beta_k} x_k^{\alpha_k - \beta_k} y... | Let $\alpha$ be a [[Definition:Multiindex|multiindex]], [[Definition:Indexing Set|indexed]] by $\set {1, \ldots, n}$ such that $\alpha_j \ge 0$ for $j = 1, \ldots, n$.
Let $x = \tuple {x_1, \ldots, x_n}$ and $y = \tuple {y_1, \ldots, y_n}$ be [[Definition:Ordered Tuple|ordered tuples]] of [[Definition:Real Number|real... | First of all, by definition of [[Definition:Power (Algebra)/Multiindices|multiindexed powers]]:
:$\ds \paren {x + y}^\alpha = \prod_{k \mathop = 1}^n \paren {x_k + y_k}^{\alpha_k}$
Then:
{{begin-eqn}}
{{eqn | l = \paren {x + y}^\alpha
| r = \prod_{k \mathop = 1}^n \sum_{\beta_k \mathop = 0}^{\alpha_k} \dbinom ... | Binomial Theorem/Multiindex | https://proofwiki.org/wiki/Binomial_Theorem/Multiindex | https://proofwiki.org/wiki/Binomial_Theorem/Multiindex | [
"Binomial Theorem"
] | [
"Definition:Multiindex",
"Definition:Indexing Set",
"Definition:Ordered Tuple",
"Definition:Real Number",
"Definition:Binomial Coefficient/Multiindices"
] | [
"Definition:Power (Algebra)/Multiindices",
"Binomial Theorem/Integral Index",
"Definition:Multiplication",
"Category:Binomial Theorem"
] |
proofwiki-6252 | Continuous Image of Path-Connected Set is Path-Connected | Let $\struct {T_1, \tau_1}, \struct {T_2, \tau_2}$ be topological spaces.
Let $f: T_1 \to T_2$ be a continuous mapping.
Let $S \subseteq T_1$ be a subset of $T_1$.
Let $S$ be path-connected in $\struct {T_1, \tau_1}$.
Then $f \sqbrk S$ is path-connected in $\struct {T_2, \tau_2}$. | Let $\map f s, \map f {s'} \in f \sqbrk S$, for some $s, s' \in S$.
Let $\mathbb I$ denote the closed unit interval:
:$\mathbb I = \closedint 0 1$
Let $p: \mathbb I \to S$ be a continuous mapping such that:
:$\map p 0 = s, \map p 1 = s'$
Such a $p$ exists since $S$ is path-connected in $\struct {T_1, \tau_1}$.
Now defi... | Let $\struct {T_1, \tau_1}, \struct {T_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $f: T_1 \to T_2$ be a [[Definition:Continuous Mapping (Topology)|continuous mapping]].
Let $S \subseteq T_1$ be a [[Definition:Subset|subset]] of $T_1$.
Let $S$ be [[Definition:Path-Connected Set|path-conne... | Let $\map f s, \map f {s'} \in f \sqbrk S$, for some $s, s' \in S$.
Let $\mathbb I$ denote the [[Definition:Closed Unit Interval|closed unit interval]]:
:$\mathbb I = \closedint 0 1$
Let $p: \mathbb I \to S$ be a [[Definition:Continuous Mapping (Topology)|continuous mapping]] such that:
:$\map p 0 = s, \map p 1 = s'... | Continuous Image of Path-Connected Set is Path-Connected | https://proofwiki.org/wiki/Continuous_Image_of_Path-Connected_Set_is_Path-Connected | https://proofwiki.org/wiki/Continuous_Image_of_Path-Connected_Set_is_Path-Connected | [
"Path-Connected Sets"
] | [
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)",
"Definition:Subset",
"Definition:Path-Connected/Set",
"Definition:Path-Connected/Set"
] | [
"Definition:Real Interval/Unit Interval/Closed",
"Definition:Continuous Mapping (Topology)",
"Definition:Path-Connected/Set",
"Composite of Continuous Mappings is Continuous",
"Definition:Continuous Mapping (Topology)",
"Definition:Path (Topology)"
] |
proofwiki-6253 | Leibniz's Rule/One Variable | Let $f$ and $g$ be real functions defined on the open interval $I$.
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $x \in I$ be a point in $I$ at which both $f$ and $g$ are $n$ times differentiable.
Then:
:$\ds \paren {\map f x \map g x}^{\paren n} = \sum_{k \mathop = 0}^n \binom n k \map {f^{\paren k} } x \... | Proof by induction: | Let $f$ and $g$ be [[Definition:Real Function|real functions]] defined on the [[Definition:Open Real Interval|open interval]] $I$.
Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $x \in I$ be a point in $I$ at which both $f$ and $g$ are [[Definition:Nth Derivative|$n... | Proof by [[Principle of Mathematical Induction|induction]]: | Leibniz's Rule/One Variable | https://proofwiki.org/wiki/Leibniz's_Rule/One_Variable | https://proofwiki.org/wiki/Leibniz's_Rule/One_Variable | [
"Leibniz's Rule"
] | [
"Definition:Real Function",
"Definition:Real Interval/Open",
"Definition:Strictly Positive/Integer",
"Definition:Derivative/Higher Derivatives/Higher Order",
"Definition:Derivative/Higher Derivatives/Order of Derivative"
] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-6254 | Leibniz's Rule/Real Valued Functions | Let $f, g : \R^n \to \R$ be real valued functions, $k$ times differentiable on some open set $U \subseteq \R^n$.
Let $\alpha = \tuple {\alpha_1, \ldots, \alpha_n}$ be a multiindex indexed by $\set {1, \ldots, n}$ with $\size \alpha \le k$.
For $i \in \set {1, \ldots, n}$, let $\partial_i$ denote the partial derivative:... | First, inserting the definitions, the statement of the theorem reads:
:$\ds \map {\partial_1^{\alpha_1} \partial_2^{\alpha_2} \cdots \partial_n^{\alpha_n} } {f g} = \sum_{\beta_1 = 0}^{\alpha_1} \cdots \sum_{\beta_n \mathop = 0}^{\alpha_n} \binom {\alpha_1} {\beta_1} \cdots \binom {\alpha_n} {\beta_n} \paren {\partial_... | Let $f, g : \R^n \to \R$ be [[Definition:Real-Valued Function|real valued functions]], [[Definition:Nth Derivative|$k$ times differentiable]] on some [[Definition:Open Set (Real Analysis)|open set]] $U \subseteq \R^n$.
Let $\alpha = \tuple {\alpha_1, \ldots, \alpha_n}$ be a [[Definition:Multiindex|multiindex]] [[Defin... | First, inserting the definitions, the statement of the theorem reads:
:$\ds \map {\partial_1^{\alpha_1} \partial_2^{\alpha_2} \cdots \partial_n^{\alpha_n} } {f g} = \sum_{\beta_1 = 0}^{\alpha_1} \cdots \sum_{\beta_n \mathop = 0}^{\alpha_n} \binom {\alpha_1} {\beta_1} \cdots \binom {\alpha_n} {\beta_n} \paren {\partial... | Leibniz's Rule/Real Valued Functions | https://proofwiki.org/wiki/Leibniz's_Rule/Real_Valued_Functions | https://proofwiki.org/wiki/Leibniz's_Rule/Real_Valued_Functions | [
"Leibniz's Rule"
] | [
"Definition:Real-Valued Function",
"Definition:Derivative/Higher Derivatives/Higher Order",
"Definition:Open Set/Real Analysis",
"Definition:Multiindex",
"Definition:Indexing Set",
"Definition:Partial Derivative",
"Definition:Partial Differential Operator",
"Definition:Function"
] | [] |
proofwiki-6255 | Open Sets of Double Pointed Topology | Let $\struct {S, \tau_S}$ be a topological space.
Let $D$ be a doubleton endowed with the indiscrete topology.
Let $\struct {S \times D, \tau}$ be the double pointed topology on $S$.
Then $X \subseteq S \times D$ is open in $\tau$ {{iff}} for some $U \in \tau$:
:$X = U \times D$ | By definition, $\tau$ is the product topology on $X \times D$.
That is, $\tau$ has as a basis sets of the form:
:$U \times V$
with $U \in \tau$ and $V$ open in $D$.
Since $D$ is endowed with the indiscrete topology, either $V = \O$ or $V = D$.
In the former case, by Cartesian Product is Empty iff Factor is Empty, $U \t... | Let $\struct {S, \tau_S}$ be a [[Definition:Topological Space|topological space]].
Let $D$ be a [[Definition:Doubleton|doubleton]] endowed with the [[Definition:Indiscrete Topology|indiscrete topology]].
Let $\struct {S \times D, \tau}$ be the [[Definition:Double Pointed Topology|double pointed topology]] on $S$.
T... | By definition, $\tau$ is the [[Definition:Product Topology|product topology]] on $X \times D$.
That is, $\tau$ has as a [[Definition:Basis (Topology)|basis]] sets of the form:
:$U \times V$
with $U \in \tau$ and $V$ [[Definition:Open Set (Topology)|open]] in $D$.
Since $D$ is endowed with the [[Definition:Indiscre... | Open Sets of Double Pointed Topology | https://proofwiki.org/wiki/Open_Sets_of_Double_Pointed_Topology | https://proofwiki.org/wiki/Open_Sets_of_Double_Pointed_Topology | [
"Double Pointed Topologies"
] | [
"Definition:Topological Space",
"Definition:Doubleton",
"Definition:Indiscrete Topology",
"Definition:Double Pointed Topology",
"Definition:Open Set/Topology"
] | [
"Definition:Product Topology",
"Definition:Basis (Topology)",
"Definition:Open Set/Topology",
"Definition:Indiscrete Topology",
"Cartesian Product is Empty iff Factor is Empty",
"Definition:Basis (Topology)",
"Definition:Open Set/Topology",
"Definition:Topology",
"Category:Double Pointed Topologies"... |
proofwiki-6256 | Open Sets of Double Pointed Topology/Corollary | A subset $X \subseteq S \times D$ is closed in $\tau$ {{iff}} for some closed set $C$ of $\tau$:
:$X = C \times D$ | By definition, $X$ is closed {{iff}} its complement $\map \complement X$ is open.
By Open Sets of Double Pointed Topology, it follows by that for some $U \in \tau$:
:$\map \complement X = U \times D$
Then by Cartesian Product Distributes over Set Difference and Complement of Complement, we have that:
{{improve|rather, ... | A [[Definition:Subset|subset]] $X \subseteq S \times D$ is [[Definition:Closed Set (Topology)|closed]] in $\tau$ {{iff}} for some [[Definition:Closed Set (Topology)|closed set]] $C$ of $\tau$:
:$X = C \times D$ | By definition, $X$ is [[Definition:Closed Set (Topology)|closed]] {{iff}} its [[Definition:Complement|complement]] $\map \complement X$ is [[Definition:Open Set (Topology)|open]].
By [[Open Sets of Double Pointed Topology]], it follows by that for some $U \in \tau$:
:$\map \complement X = U \times D$
Then by [[Carte... | Open Sets of Double Pointed Topology/Corollary | https://proofwiki.org/wiki/Open_Sets_of_Double_Pointed_Topology/Corollary | https://proofwiki.org/wiki/Open_Sets_of_Double_Pointed_Topology/Corollary | [
"Double Pointed Topologies"
] | [
"Definition:Subset",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology"
] | [
"Definition:Closed Set/Topology",
"Definition:Complement",
"Definition:Open Set/Topology",
"Open Sets of Double Pointed Topology",
"Cartesian Product Distributes over Set Difference",
"Complement of Complement",
"Cartesian Product Distributes over Complement",
"Definition:Closed Set/Topology",
"Cate... |
proofwiki-6257 | Open Set Less One Point is Open/Corollary | Let $S = \set {\alpha_1, \alpha_2, \ldots, \alpha_n} \subseteq U$ be a finite set of points in $U$.
Then $U \setminus S$ is open in $M$. | Follows directly from Open Set Less One Point is Open and Finite Intersection of Open Sets of Metric Space is Open.
Let:
:$U_1 = U \setminus \set {\alpha_1}, U_2 = U \setminus \set {\alpha_2}, \ldots, U_n = U \setminus \set {\alpha_n}$
From the above, $U_1, U_2, \ldots, U_n$ are all open in $M$.
From Finite Intersectio... | Let $S = \set {\alpha_1, \alpha_2, \ldots, \alpha_n} \subseteq U$ be a [[Definition:Finite Set|finite set]] of points in $U$.
Then $U \setminus S$ is [[Definition:Open Set (Metric Space)|open]] in $M$. | Follows directly from [[Open Set Less One Point is Open]] and [[Finite Intersection of Open Sets of Metric Space is Open]].
Let:
:$U_1 = U \setminus \set {\alpha_1}, U_2 = U \setminus \set {\alpha_2}, \ldots, U_n = U \setminus \set {\alpha_n}$
From the above, $U_1, U_2, \ldots, U_n$ are all [[Definition:Open Set (Met... | Open Set Less One Point is Open/Corollary | https://proofwiki.org/wiki/Open_Set_Less_One_Point_is_Open/Corollary | https://proofwiki.org/wiki/Open_Set_Less_One_Point_is_Open/Corollary | [
"Open Set Less One Point is Open"
] | [
"Definition:Finite Set",
"Definition:Open Set/Metric Space"
] | [
"Open Set Less One Point is Open",
"Finite Intersection of Open Sets of Metric Space is Open",
"Definition:Open Set/Metric Space",
"Finite Intersection of Open Sets of Metric Space is Open",
"Definition:Open Set/Metric Space",
"Category:Open Set Less One Point is Open"
] |
proofwiki-6258 | Closed Real Interval is Neighborhood Except at Endpoints | Let $\R$ be the real number line considered as an Euclidean space.
Let $\closedint a b \subset \R$ be a closed interval of $\R$.
Then $\closedint a b$ is a neighborhood of all of its points except $a$ and $b$. | Let $a < c < b$.
Let $\epsilon < \map \min {b - c, c - a}$.
From the definition of positive it follows that $\epsilon \in \R_{>0}$.
Let $\map {B_\epsilon} c = \openint {c - \epsilon} {c + \epsilon}$ be the open $\epsilon$-ball of $c$.
We have that $c + \epsilon < b$ and $a < c - \epsilon$.
Thus:
:$\map {B_\epsilon} c \... | Let $\R$ be the [[Definition:Real Number Line|real number line]] considered as an [[Definition:Euclidean Space|Euclidean space]].
Let $\closedint a b \subset \R$ be a [[Definition:Closed Real Interval|closed interval]] of $\R$.
Then $\closedint a b$ is a [[Definition:Neighborhood (Metric Space)|neighborhood]] of all... | Let $a < c < b$.
Let $\epsilon < \map \min {b - c, c - a}$.
From the definition of [[Definition:Positive Real Number|positive]] it follows that $\epsilon \in \R_{>0}$.
Let $\map {B_\epsilon} c = \openint {c - \epsilon} {c + \epsilon}$ be the [[Definition:Open Ball|open $\epsilon$-ball]] of $c$.
We have that $c + \... | Closed Real Interval is Neighborhood Except at Endpoints | https://proofwiki.org/wiki/Closed_Real_Interval_is_Neighborhood_Except_at_Endpoints | https://proofwiki.org/wiki/Closed_Real_Interval_is_Neighborhood_Except_at_Endpoints | [
"Real Intervals",
"Neighborhoods"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space",
"Definition:Real Interval/Closed",
"Definition:Neighborhood (Metric Space)"
] | [
"Definition:Positive/Real Number",
"Definition:Open Ball",
"Definition:Neighborhood (Metric Space)",
"Definition:Neighborhood (Metric Space)"
] |
proofwiki-6259 | Open Real Interval is Open Set | Let $\R$ be the real number line considered as an Euclidean space.
Let $\openint a b \subset \R$ be an open interval of $\R$.
Then $\openint a b$ is an open set of $\R$. | Let $c \in \R$ such that $a < c < b$.
Let $\epsilon = \min \set {b - c, c - a}$.
From the definition of positive it follows that $\epsilon \in \R_{>0}$.
Let $\map {B_\epsilon} c = \openint {c - \epsilon} {c + \epsilon}$ be the open $\epsilon$-ball of $c$.
Let $y \in \map {B_\epsilon} c$.
Then:
:$\size {c - y} < \epsilo... | Let $\R$ be the [[Definition:Real Number Line|real number line]] considered as an [[Definition:Euclidean Space|Euclidean space]].
Let $\openint a b \subset \R$ be an [[Definition:Open Real Interval|open interval]] of $\R$.
Then $\openint a b$ is an [[Definition:Open Set (Metric Space)|open set]] of $\R$. | Let $c \in \R$ such that $a < c < b$.
Let $\epsilon = \min \set {b - c, c - a}$.
From the definition of [[Definition:Positive Real Number|positive]] it follows that $\epsilon \in \R_{>0}$.
Let $\map {B_\epsilon} c = \openint {c - \epsilon} {c + \epsilon}$ be the [[Definition:Open Ball of Metric Space|open $\epsilon... | Open Real Interval is Open Set | https://proofwiki.org/wiki/Open_Real_Interval_is_Open_Set | https://proofwiki.org/wiki/Open_Real_Interval_is_Open_Set | [
"Real Intervals",
"Open Sets"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space",
"Definition:Real Interval/Open",
"Definition:Open Set/Metric Space"
] | [
"Definition:Positive/Real Number",
"Definition:Open Ball",
"Definition:Neighborhood (Metric Space)",
"Definition:Open Set/Metric Space"
] |
proofwiki-6260 | Closed Real Interval is not Open Set | Let $\R$ be the real number line considered as a Euclidean space.
Let $\closedint a b \subset \R$ be a closed interval of $\R$.
Then $\closedint a b$ is not an open set of $\R$. | From Closed Real Interval is Neighborhood Except at Endpoints, $a$ and $b$ have no open $\epsilon$-ball lying entirely in $\closedint a b$.
The result follows by definition of open set.
{{qed}} | Let $\R$ be the [[Definition:Real Number Line|real number line]] considered as a [[Definition:Euclidean Space|Euclidean space]].
Let $\closedint a b \subset \R$ be a [[Definition:Closed Real Interval|closed interval]] of $\R$.
Then $\closedint a b$ is not an [[Definition:Open Set (Metric Space)|open set]] of $\R$. | From [[Closed Real Interval is Neighborhood Except at Endpoints]], $a$ and $b$ have no [[Definition:Open Ball|open $\epsilon$-ball]] lying entirely in $\closedint a b$.
The result follows by definition of [[Definition:Open Set (Metric Space)|open set]].
{{qed}} | Closed Real Interval is not Open Set | https://proofwiki.org/wiki/Closed_Real_Interval_is_not_Open_Set | https://proofwiki.org/wiki/Closed_Real_Interval_is_not_Open_Set | [
"Real Intervals",
"Open Sets"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space",
"Definition:Real Interval/Closed",
"Definition:Open Set/Metric Space"
] | [
"Closed Real Interval is Neighborhood Except at Endpoints",
"Definition:Open Ball",
"Definition:Open Set/Metric Space"
] |
proofwiki-6261 | Open Set may not be Open Ball | Let $M = \struct {A, d}$ be a metric space.
Let $U \subseteq M$ be an open set of $M$.
Then it is not necessarily the case that $U$ is an open ball of some $x \in A$. | Consider the Euclidean space $\R^2$.
Let:
:$U \subseteq \R^2: U = \set {\tuple {x_1, x_2}: a < x_1 < b, c < x_2 < d}$
Let $x = \tuple {x_1, x_2} \in U$.
Then $\map {B_\epsilon} x \subseteq U$ when $\epsilon = \min \set {x_1 - a, b - x_1, x_2 - c, d - x_2}$:
:310px
So by definition, $U$ is open in $M$.
However, $U$ is n... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $U \subseteq M$ be an [[Definition:Open Set of Metric Space|open set]] of $M$.
Then it is not necessarily the case that $U$ is an [[Definition:Open Ball of Metric Space|open ball]] of some $x \in A$. | Consider the [[Definition:Euclidean Space|Euclidean space]] $\R^2$.
Let:
:$U \subseteq \R^2: U = \set {\tuple {x_1, x_2}: a < x_1 < b, c < x_2 < d}$
Let $x = \tuple {x_1, x_2} \in U$.
Then $\map {B_\epsilon} x \subseteq U$ when $\epsilon = \min \set {x_1 - a, b - x_1, x_2 - c, d - x_2}$:
:[[File:NeighborhoodInOpenS... | Open Set may not be Open Ball | https://proofwiki.org/wiki/Open_Set_may_not_be_Open_Ball | https://proofwiki.org/wiki/Open_Set_may_not_be_Open_Ball | [
"Open Sets (Metric Spaces)",
"Open Balls"
] | [
"Definition:Metric Space",
"Definition:Open Set/Metric Space",
"Definition:Open Ball"
] | [
"Definition:Euclidean Space",
"File:NeighborhoodInOpenSet.png",
"Definition:Open Set/Metric Space",
"Definition:Open Ball"
] |
proofwiki-6262 | Quotient Mapping is Coequalizer | Let $\mathbf{Set}$ be the category of sets.
Let $S$ be a Set.
Let $\RR \subseteq S \times S$ be an equivalence relation on $S$.
Let $r_1, r_2: \RR \to S$ be the projections corresponding to the inclusion mapping $\RR \hookrightarrow S \times S$.
Let $q: S \to S / \RR$ be the quotient mapping induced by $\RR$.
Then $q$ ... | Let $f: S \to T$ be a mapping as in the following commutative diagram:
$\quad\quad \begin{xy}\xymatrix{
\RR
\ar[r]<2pt>^*{r_1}
\ar[r]<-2pt>_*{r_2}
&
S
\ar[r]^*{q}
\ar[rd]_*{f}
&
S / \RR
\ar@{.>}[d]^*{\bar f}
\\
& &
T
}\end{xy}$
This translates to, for $s_1, s_2 \in S$ with $s_1 \RR s_2$:
:$\map {f \circ r_1} {s_1, s_... | Let $\mathbf{Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $S$ be a [[Definition:Set|Set]].
Let $\RR \subseteq S \times S$ be an [[Definition:Equivalence Relation|equivalence relation]] on $S$.
Let $r_1, r_2: \RR \to S$ be the [[Definition:Projection (Category Theory)|projections]] corresponding... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]] as in the following [[Definition:Commutative Diagram|commutative diagram]]:
$\quad\quad \begin{xy}\xymatrix{
\RR
\ar[r]<2pt>^*{r_1}
\ar[r]<-2pt>_*{r_2}
&
S
\ar[r]^*{q}
\ar[rd]_*{f}
&
S / \RR
\ar@{.>}[d]^*{\bar f}
\\
& &
T
}\end{xy}$
This translates to, for $s_1, ... | Quotient Mapping is Coequalizer | https://proofwiki.org/wiki/Quotient_Mapping_is_Coequalizer | https://proofwiki.org/wiki/Quotient_Mapping_is_Coequalizer | [
"Category of Sets",
"Quotient Mappings"
] | [
"Definition:Category of Sets",
"Definition:Set",
"Definition:Equivalence Relation",
"Definition:Product (Category Theory)/Projection",
"Definition:Inclusion Mapping",
"Definition:Quotient Mapping",
"Definition:Coequalizer"
] | [
"Definition:Mapping",
"Definition:Commutative Diagram",
"Definition:Well-Defined/Mapping",
"Definition:Unique",
"Definition:Coequalizer"
] |
proofwiki-6263 | Closed Real Interval is Closed Set | Let $\R$ be the real number line considered as an Euclidean space.
Let $\closedint a b \subset \R$ be a closed interval of $\R$.
Then $\closedint a b$ is a closed set of $\R$. | {{begin-eqn}}
{{eqn | l = \closedint a b
| r = \set {x \in \R: x \ge a \land x \le b}
| c = {{Defof|Closed Real Interval}}
}}
{{eqn | ll= \leadsto
| l = \R \setminus \closedint a b
| r = \R \setminus \set {x \in \R: x \ge a \land x \le b}
| c =
}}
{{eqn | r = \set {x \in \R: x < a \lor x ... | Let $\R$ be the [[Definition:Real Number Line|real number line]] considered as an [[Definition:Euclidean Space|Euclidean space]].
Let $\closedint a b \subset \R$ be a [[Definition:Closed Real Interval|closed interval]] of $\R$.
Then $\closedint a b$ is a [[Definition:Closed Set (Metric Space)|closed set]] of $\R$. | {{begin-eqn}}
{{eqn | l = \closedint a b
| r = \set {x \in \R: x \ge a \land x \le b}
| c = {{Defof|Closed Real Interval}}
}}
{{eqn | ll= \leadsto
| l = \R \setminus \closedint a b
| r = \R \setminus \set {x \in \R: x \ge a \land x \le b}
| c =
}}
{{eqn | r = \set {x \in \R: x < a \lor x ... | Closed Real Interval is Closed Set | https://proofwiki.org/wiki/Closed_Real_Interval_is_Closed_Set | https://proofwiki.org/wiki/Closed_Real_Interval_is_Closed_Set | [
"Metric Spaces",
"Real Intervals",
"Closed Sets"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space",
"Definition:Real Interval/Closed",
"Definition:Closed Set/Metric Space"
] | [
"De Morgan's Laws (Logic)/Disjunction of Negations",
"Definition:Open Set/Metric Space",
"Union of Open Sets of Metric Space is Open",
"Definition:Open Set/Metric Space",
"Definition:Relative Complement",
"Definition:Closed Set/Metric Space"
] |
proofwiki-6264 | Kernel of Linear Transformation is Null Space of Matrix Representation | Let $V$ and $W$ be finite dimensional vector spaces.
Let $\phi: V \to W$ be a linear transformation from $V$ to $W$.
Let $\tuple {e_1, \ldots, e_n}$ and $\tuple {f_1, \ldots, f_m}$ be ordered bases of $V$ and $W$ respectively.
Let $A$ be the matrix of $\phi$ in these bases.
Define $f: V \to \R^n$ by:
:$\ds \sum_{i \mat... | {{MissingLinks}}
{{proofread|Use of injectivity of $g$ here is implicit; but I ''think'' it's still rigorous without details: second opinion welcome}}
By the definition of the matrix $A$:
:$A \circ f = g \circ \phi$
Therefore if $x \in \map \ker \phi$ we have:
:$A \map f x = \map g {\map \phi x} = \map g 0 = 0$
This sh... | Let $V$ and $W$ be [[Definition:Finite Dimensional Vector Space|finite dimensional vector spaces]].
Let $\phi: V \to W$ be a [[Definition:Linear Transformation|linear transformation]] from $V$ to $W$.
Let $\tuple {e_1, \ldots, e_n}$ and $\tuple {f_1, \ldots, f_m}$ be [[Definition:Ordered Basis|ordered bases]] of $V$ ... | {{MissingLinks}}
{{proofread|Use of injectivity of $g$ here is implicit; but I ''think'' it's still rigorous without details: second opinion welcome}}
By the definition of the [[Definition:Matrix|matrix]] $A$:
:$A \circ f = g \circ \phi$
Therefore if $x \in \map \ker \phi$ we have:
:$A \map f x = \map g {\map \phi x}... | Kernel of Linear Transformation is Null Space of Matrix Representation | https://proofwiki.org/wiki/Kernel_of_Linear_Transformation_is_Null_Space_of_Matrix_Representation | https://proofwiki.org/wiki/Kernel_of_Linear_Transformation_is_Null_Space_of_Matrix_Representation | [
"Kernels of Linear Transformations",
"Null Spaces"
] | [
"Definition:Dimension of Vector Space/Finite",
"Definition:Linear Transformation",
"Definition:Ordered Basis",
"Definition:Relative Matrix of Linear Transformation",
"Definition:Null Space",
"Definition:Kernel of Linear Transformation",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Subse... | [
"Definition:Matrix",
"Category:Kernels of Linear Transformations",
"Category:Null Spaces"
] |
proofwiki-6265 | Half-Open Real Interval is neither Open nor Closed | Let $\R$ be the real number line considered as an Euclidean space.
Let $\hointr a b \subset \R$ be a right half-open interval of $\R$.
Then $\hointr a b$ is neither an open set nor a closed set of $\R$.
Similarly, the left half-open interval $\hointl a b \subset \R$ is neither an open set nor a closed set of $\R$. | From Half-Open Real Interval is not Open Set we have that neither $\hointr a b$ nor $\hointl a b$ is an open set of $\R$.
From Half-Open Real Interval is not Closed in Real Number Line we have that neither $\hointr a b$ nor $\hointl a b$ is a closed set of $\R$.
{{qed}} | Let $\R$ be the [[Definition:Real Number Line|real number line]] considered as an [[Definition:Euclidean Space|Euclidean space]].
Let $\hointr a b \subset \R$ be a [[Definition:Right Half-Open Real Interval|right half-open interval]] of $\R$.
Then $\hointr a b$ is neither an [[Definition:Open Set (Metric Space)|open... | From [[Half-Open Real Interval is not Open Set]] we have that neither $\hointr a b$ nor $\hointl a b$ is an [[Definition:Open Set (Metric Space)|open set]] of $\R$.
From [[Half-Open Real Interval is not Closed in Real Number Line]] we have that neither $\hointr a b$ nor $\hointl a b$ is a [[Definition:Closed Set (Metr... | Half-Open Real Interval is neither Open nor Closed | https://proofwiki.org/wiki/Half-Open_Real_Interval_is_neither_Open_nor_Closed | https://proofwiki.org/wiki/Half-Open_Real_Interval_is_neither_Open_nor_Closed | [
"Real Intervals",
"Open Sets",
"Closed Sets"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space",
"Definition:Real Interval/Half-Open/Right",
"Definition:Open Set/Metric Space",
"Definition:Closed Set/Metric Space",
"Definition:Real Interval/Half-Open/Left",
"Definition:Open Set/Metric Space",
"Definition:Closed Set/Metric Sp... | [
"Half-Open Real Interval is not Open Set",
"Definition:Open Set/Metric Space",
"Half-Open Real Interval is not Closed in Real Number Line",
"Definition:Closed Set/Metric Space"
] |
proofwiki-6266 | Compact Sets of Double Pointed Topology | Let $\struct {S, \tau_S}$ be a topological space.
Let $D$ be a doubleton endowed with the indiscrete topology.
Let $\struct {S \times D, \tau}$ be the double pointed topology on $S$.
Then $X \subseteq S \times D$ is compact in $\tau$ {{iff}} for some compact set $C$ of $\tau_S$:
:$\map {\pr_1} X = C$
where $\pr_1$ deno... | === Necessary Condition ===
Suppose that $X \subseteq S \times D$ is a compact set in $\vartheta$.
It is to be shown that $C = \map {\pr_1} X$ is compact in $\tau$.
This follows from Compactness Properties Preserved under Projection Mapping.
{{qed|lemma}} | Let $\struct {S, \tau_S}$ be a [[Definition:Topological Space|topological space]].
Let $D$ be a [[Definition:Doubleton|doubleton]] endowed with the [[Definition:Indiscrete Topology|indiscrete topology]].
Let $\struct {S \times D, \tau}$ be the [[Definition:Double Pointed Topology|double pointed topology]] on $S$.
T... | === Necessary Condition ===
Suppose that $X \subseteq S \times D$ is a [[Definition:Compact Set (Topology)|compact set]] in $\vartheta$.
It is to be shown that $C = \map {\pr_1} X$ is [[Definition:Compact Set (Topology)|compact]] in $\tau$.
This follows from [[Compactness Properties Preserved under Projection Mappin... | Compact Sets of Double Pointed Topology | https://proofwiki.org/wiki/Compact_Sets_of_Double_Pointed_Topology | https://proofwiki.org/wiki/Compact_Sets_of_Double_Pointed_Topology | [
"Double Pointed Topologies"
] | [
"Definition:Topological Space",
"Definition:Doubleton",
"Definition:Indiscrete Topology",
"Definition:Double Pointed Topology",
"Definition:Compact Topological Space/Subspace",
"Definition:Compact Topological Space/Subspace",
"Definition:Projection (Mapping Theory)/First Projection"
] | [
"Definition:Compact Topological Space/Subspace",
"Definition:Compact Topological Space/Subspace",
"Compactness Properties Preserved under Projection Mapping",
"Definition:Compact Topological Space/Subspace",
"Definition:Compact Topological Space/Subspace",
"Definition:Compact Topological Space/Subspace"
] |
proofwiki-6267 | Closure of Subset of Double Pointed Topological Space | Let $\struct {S, \tau_S}$ be a topological space.
Let $D$ be a doubleton endowed with the indiscrete topology.
Let $\struct {S \times D, \tau}$ be the double pointed topology on $S$.
Let $X \subseteq S \times D$ be a subset of $S \times D$.
Then the closure of $X$ in $\tau$ is:
:$\map \cl X = \map \cl {\map {\pr_1} X} ... | By Closed Sets of Double Pointed Topology, $\map \cl {\map {\pr_1} X} \times D$ is closed in $\tau$.
Furthermore, for $\tuple {s, d} \in X$, one has:
:$s \in \map {\pr_1} X \subseteq \map \cl {\map {\pr_1} X}$
by definition of closure.
Since also $d \in D$, we conclude that:
:$X \subseteq \map \cl {\map {\pr_1} X} \tim... | Let $\struct {S, \tau_S}$ be a [[Definition:Topological Space|topological space]].
Let $D$ be a [[Definition:Doubleton|doubleton]] endowed with the [[Definition:Indiscrete Topology|indiscrete topology]].
Let $\struct {S \times D, \tau}$ be the [[Definition:Double Pointed Topology|double pointed topology]] on $S$.
Le... | By [[Closed Sets of Double Pointed Topology]], $\map \cl {\map {\pr_1} X} \times D$ is [[Definition:Closed Set (Topology)|closed]] in $\tau$.
Furthermore, for $\tuple {s, d} \in X$, one has:
:$s \in \map {\pr_1} X \subseteq \map \cl {\map {\pr_1} X}$
by definition of [[Definition:Closure (Topology)|closure]].
Since... | Closure of Subset of Double Pointed Topological Space | https://proofwiki.org/wiki/Closure_of_Subset_of_Double_Pointed_Topological_Space | https://proofwiki.org/wiki/Closure_of_Subset_of_Double_Pointed_Topological_Space | [
"Double Pointed Topologies",
"Set Closures"
] | [
"Definition:Topological Space",
"Definition:Doubleton",
"Definition:Indiscrete Topology",
"Definition:Double Pointed Topology",
"Definition:Subset",
"Definition:Closure (Topology)",
"Definition:Projection (Mapping Theory)/First Projection"
] | [
"Open Sets of Double Pointed Topology/Corollary",
"Definition:Closed Set/Topology",
"Definition:Closure (Topology)",
"Equivalence of Definitions of Closure of Topological Subspace",
"Definition:Closed Set/Topology",
"Open Sets of Double Pointed Topology/Corollary",
"Definition:Closed Set/Topology",
"I... |
proofwiki-6268 | Interior of Subset of Double Pointed Topological Space | Let $\struct {S, \tau_S}$ be a topological space.
Let $D$ be a doubleton endowed with the indiscrete topology.
Let $\struct {S \times D, \tau}$ be the double pointed topology on $S$.
Let $X \subseteq S \times D$ be a subset of $S \times D$.
Define $A \subseteq S$ by:
:$A := \set {s \in S: \paren {\forall d \in D: \tupl... | By Open Sets of Double Pointed Topology, $X^\circ$ must be of the form:
:$X^\circ = U \times D$
with $U$ open in $\tau_S$.
By Set Interior is Largest Open Set, we have for any open set $U' \times D$ of $\tau$ that:
:$U' \times D \subseteq X \iff U' \times D \subseteq X^\circ = U \times D$
By Cartesian Product of Subset... | Let $\struct {S, \tau_S}$ be a [[Definition:Topological Space|topological space]].
Let $D$ be a [[Definition:Doubleton|doubleton]] endowed with the [[Definition:Indiscrete Topology|indiscrete topology]].
Let $\struct {S \times D, \tau}$ be the [[Definition:Double Pointed Topology|double pointed topology]] on $S$.
Le... | By [[Open Sets of Double Pointed Topology]], $X^\circ$ must be of the form:
:$X^\circ = U \times D$
with $U$ [[Definition:Open Set (Topology)|open]] in $\tau_S$.
By [[Set Interior is Largest Open Set]], we have for any [[Definition:Open Set (Topology)|open set]] $U' \times D$ of $\tau$ that:
:$U' \times D \subsete... | Interior of Subset of Double Pointed Topological Space | https://proofwiki.org/wiki/Interior_of_Subset_of_Double_Pointed_Topological_Space | https://proofwiki.org/wiki/Interior_of_Subset_of_Double_Pointed_Topological_Space | [
"Double Pointed Topologies",
"Examples of Set Interiors"
] | [
"Definition:Topological Space",
"Definition:Doubleton",
"Definition:Indiscrete Topology",
"Definition:Double Pointed Topology",
"Definition:Subset",
"Definition:Interior (Topology)"
] | [
"Open Sets of Double Pointed Topology",
"Definition:Open Set/Topology",
"Equivalence of Definitions of Interior (Topology)",
"Definition:Open Set/Topology",
"Cartesian Product of Subsets",
"Definition:Open Set/Topology",
"Definition:Non-Empty Set",
"Definition:Open Set/Topology",
"Equivalence of Def... |
proofwiki-6269 | Cartesian Product with Complement | Let $S$ and $T$ be sets.
Let $A \subseteq S$ and $B \subseteq T$ be subsets of $S$ and $T$, respectively.
Let $\relcomp S A$ denote the relative complement of $A$ in $S$.
Then:
{{begin-eqn}}
{{eqn | l = \relcomp S A \times T
| r = \relcomp {S \times T} {A \times T}
}}
{{eqn | l = S \times \relcomp T B
| r ... | By definition of relative complement we have:
:$\relcomp S A = S \setminus A$
where $S \setminus A$ denotes set difference.
By Cartesian Product Distributes over Set Difference, we have:
:$\paren {S \setminus A} \times T = \paren {S \times T} \setminus \paren {A \times T}$
and the latter equals $\relcomp {S \times T} {... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $A \subseteq S$ and $B \subseteq T$ be [[Definition:Subset|subsets]] of $S$ and $T$, respectively.
Let $\relcomp S A$ denote the [[Definition:Relative Complement|relative complement]] of $A$ in $S$.
Then:
{{begin-eqn}}
{{eqn | l = \relcomp S A \times T
| r = \... | By definition of [[Definition:Relative Complement|relative complement]] we have:
:$\relcomp S A = S \setminus A$
where $S \setminus A$ denotes [[Definition:Set Difference|set difference]].
By [[Cartesian Product Distributes over Set Difference]], we have:
:$\paren {S \setminus A} \times T = \paren {S \times T} \set... | Cartesian Product with Complement | https://proofwiki.org/wiki/Cartesian_Product_with_Complement | https://proofwiki.org/wiki/Cartesian_Product_with_Complement | [
"Cartesian Product",
"Set Complement"
] | [
"Definition:Set",
"Definition:Subset",
"Definition:Relative Complement"
] | [
"Definition:Relative Complement",
"Definition:Set Difference",
"Cartesian Product Distributes over Set Difference",
"Category:Cartesian Product",
"Category:Set Complement"
] |
proofwiki-6270 | Projection of Complement Contains Complement of Projection | Let $S$ and $T$ be non-empty sets.
Let $X \subseteq S \times T$ be a subset of the Cartesian product $S \times T$.
Denote with $\pr_1, \pr_2$ and $\complement$ the first and second projections, and the complement operation, respectively.
Then:
{{begin-eqn}}
{{eqn|l = \map \complement {\map {\pr_1} X}
|o = \subsete... | Let $s \in S$.
Then:
{{begin-eqn}}
{{eqn|l = s
|o = \in
|r = \map \complement {\map {\pr_1} X}
}}
{{eqn|ll= \leadstoandfrom
|l = s
|o = \notin
|r = \map {\pr_1} X
|c = {{Defof|Set Complement}}
}}
{{eqn|ll= \leadstoandfrom
|q = \forall t \in T
|l = \tuple {s, t}
|o = \notin
... | Let $S$ and $T$ be [[Definition:Non-Empty Set|non-empty sets]].
Let $X \subseteq S \times T$ be a [[Definition:Subset|subset]] of the [[Definition:Cartesian Product|Cartesian product]] $S \times T$.
Denote with $\pr_1, \pr_2$ and $\complement$ the [[Definition:First Projection|first]] and [[Definition:Second Projecti... | Let $s \in S$.
Then:
{{begin-eqn}}
{{eqn|l = s
|o = \in
|r = \map \complement {\map {\pr_1} X}
}}
{{eqn|ll= \leadstoandfrom
|l = s
|o = \notin
|r = \map {\pr_1} X
|c = {{Defof|Set Complement}}
}}
{{eqn|ll= \leadstoandfrom
|q = \forall t \in T
|l = \tuple {s, t}
|o = \notin... | Projection of Complement Contains Complement of Projection | https://proofwiki.org/wiki/Projection_of_Complement_Contains_Complement_of_Projection | https://proofwiki.org/wiki/Projection_of_Complement_Contains_Complement_of_Projection | [
"Projections",
"Set Complement"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Cartesian Product",
"Definition:Projection (Mapping Theory)/First Projection",
"Definition:Projection (Mapping Theory)/Second Projection",
"Definition:Set Complement"
] | [
"Universal Instantiation",
"Existential Generalisation",
"Definition:Subset",
"Category:Projections",
"Category:Set Complement"
] |
proofwiki-6271 | Interior of Subset | Let $\left({S, \tau}\right)$ be a topological space.
Let $X$ and $Y$ be subsets of $S$, and suppose that $X \subseteq Y$.
Then:
:$X^\circ \subseteq Y^\circ$
where $X^\circ$ denotes the interior of $X$. | By definition of interior, $X^\circ$ is open in $\tau$, and:
:$Y^\circ \subseteq Y$
Hence, by Subset Relation is Transitive:
:$X^\circ \subseteq Y$
is an open set contained in $Y$.
The result follows by Set Interior is Largest Open Set.
{{qed}}
Category:Subsets
Category:Set Interiors
rh05xi8umelkm4d6icp3t0sg58yibbu | Let $\left({S, \tau}\right)$ be a [[Definition:Topological Space|topological space]].
Let $X$ and $Y$ be [[Definition:Subset|subsets]] of $S$, and suppose that $X \subseteq Y$.
Then:
:$X^\circ \subseteq Y^\circ$
where $X^\circ$ denotes the [[Definition:Interior (Topology)|interior]] of $X$. | By definition of [[Definition:Interior (Topology)|interior]], $X^\circ$ is [[Definition:Open Set (Topology)|open]] in $\tau$, and:
:$Y^\circ \subseteq Y$
Hence, by [[Subset Relation is Transitive]]:
:$X^\circ \subseteq Y$
is an [[Definition:Open Set (Topology)|open set]] contained in $Y$.
The result follows by [[... | Interior of Subset | https://proofwiki.org/wiki/Interior_of_Subset | https://proofwiki.org/wiki/Interior_of_Subset | [
"Subsets",
"Set Interiors"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Interior (Topology)"
] | [
"Definition:Interior (Topology)",
"Definition:Open Set/Topology",
"Subset Relation is Transitive",
"Definition:Open Set/Topology",
"Equivalence of Definitions of Interior (Topology)",
"Category:Subsets",
"Category:Set Interiors"
] |
proofwiki-6272 | Real Interval is Bounded in Real Numbers | Let $\R$ be the real number line considered as an Euclidean space.
Let $a, b \in \R$.
Let $\II$ be one of the following real intervals:
{{begin-eqn}}
{{eqn | l = \openint a b
| o = :=
| r = \set {x \in \R: a < x < b}
| c = Open Real Interval
}}
{{eqn | l = \hointr a b
| o = :=
| r = \set {... | Consider the open $\epsilon$-ball $\map {B_\epsilon} a$ where $\epsilon = b + 1 - a$.
As $b \ge a$ we have that $b + 1 > a$ and so $\epsilon > 0$.
Let $x \in \II$.
Then, whatever type of real interval $\II$ actually is, $z \ge a$ and $x \le b$.
As $\epsilon > 0$ it follows that $x > a - \epsilon$.
Also:
{{begin-eqn}}
{... | Let $\R$ be the [[Definition:Real Number Line|real number line]] considered as an [[Definition:Euclidean Space|Euclidean space]].
Let $a, b \in \R$.
Let $\II$ be one of the following [[Definition:Real Interval|real intervals]]:
{{begin-eqn}}
{{eqn | l = \openint a b
| o = :=
| r = \set {x \in \R: a < x <... | Consider the [[Definition:Open Ball|open $\epsilon$-ball]] $\map {B_\epsilon} a$ where $\epsilon = b + 1 - a$.
As $b \ge a$ we have that $b + 1 > a$ and so $\epsilon > 0$.
Let $x \in \II$.
Then, whatever type of [[Definition:Real Interval|real interval]] $\II$ actually is, $z \ge a$ and $x \le b$.
As $\epsilon > 0... | Real Interval is Bounded in Real Numbers | https://proofwiki.org/wiki/Real_Interval_is_Bounded_in_Real_Numbers | https://proofwiki.org/wiki/Real_Interval_is_Bounded_in_Real_Numbers | [
"Real Intervals"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space",
"Definition:Real Interval",
"Definition:Real Interval/Open",
"Definition:Real Interval/Half-Open/Right",
"Definition:Real Interval/Half-Open/Left",
"Definition:Real Interval/Closed",
"Definition:Bounded Metric Space"
] | [
"Definition:Open Ball",
"Definition:Real Interval",
"Definition:Bounded Metric Space"
] |
proofwiki-6273 | Set of Integers is not Bounded | Let $\R$ be the real number line considered as an Euclidean space.
The set $\Z$ of integers is not bounded in $\R$. | Let $a \in \R$.
Let $K \in \R_{>0}$.
Consider the open $K$-ball $\map {B_K} a$.
By the Axiom of Archimedes there exists $n \in \N$ such that $n > a + K$.
As $\N \subseteq \Z$:
:$\exists n \in \Z: a + K < n$
and so:
:$n \notin \map {B_K} a$
As this applies whatever $a$ and $K$ are, it follows that there is no $\map {B_K... | Let $\R$ be the [[Definition:Real Number Line|real number line]] considered as an [[Definition:Euclidean Space|Euclidean space]].
The set $\Z$ of [[Definition:Integer|integers]] is not [[Definition:Bounded Space|bounded]] in $\R$. | Let $a \in \R$.
Let $K \in \R_{>0}$.
Consider the [[Definition:Open Ball|open $K$-ball]] $\map {B_K} a$.
By the [[Axiom of Archimedes]] there exists $n \in \N$ such that $n > a + K$.
As $\N \subseteq \Z$:
:$\exists n \in \Z: a + K < n$
and so:
:$n \notin \map {B_K} a$
As this applies whatever $a$ and $K$ are, it f... | Set of Integers is not Bounded | https://proofwiki.org/wiki/Set_of_Integers_is_not_Bounded | https://proofwiki.org/wiki/Set_of_Integers_is_not_Bounded | [
"Real Analysis",
"Boundedness"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space",
"Definition:Integer",
"Definition:Bounded Metric Space"
] | [
"Definition:Open Ball",
"Axiom of Archimedes",
"Definition:Integer",
"Definition:Bounded Metric Space"
] |
proofwiki-6274 | Open Real Interval is not Closed Set/Corollary | Let:
:$I_a = \openint \gets a$
:$I_b = \openint b \to$
be unbounded open real intervals.
Then neither $I_a$ nor $I_b$ are closed sets of $\R$. | Consider the relative complement of $I_a$ in $\R$:
:$J = \relcomp \R I = \R \setminus I = \hointr a \to$
Let $\epsilon \in \R_{>0}$.
Consider the open $\epsilon$-ball $\map {B_\epsilon} a$.
Whatever the value of $\epsilon$ is, $a - \epsilon$ is not in $\map {B_\epsilon} a$.
So, by definition, $J$ is not an open set of ... | Let:
:$I_a = \openint \gets a$
:$I_b = \openint b \to$
be [[Definition:Unbounded Open Real Interval|unbounded open real intervals]].
Then neither $I_a$ nor $I_b$ are [[Definition:Closed Set (Metric Space)|closed sets]] of $\R$. | Consider the [[Definition:Relative Complement|relative complement]] of $I_a$ in $\R$:
:$J = \relcomp \R I = \R \setminus I = \hointr a \to$
Let $\epsilon \in \R_{>0}$.
Consider the [[Definition:Open Ball|open $\epsilon$-ball]] $\map {B_\epsilon} a$.
Whatever the value of $\epsilon$ is, $a - \epsilon$ is not in $\map... | Open Real Interval is not Closed Set/Corollary | https://proofwiki.org/wiki/Open_Real_Interval_is_not_Closed_Set/Corollary | https://proofwiki.org/wiki/Open_Real_Interval_is_not_Closed_Set/Corollary | [
"Real Intervals",
"Real Number Line with Euclidean Metric"
] | [
"Definition:Real Interval/Unbounded Open",
"Definition:Closed Set/Metric Space"
] | [
"Definition:Relative Complement",
"Definition:Open Ball",
"Definition:Open Set/Metric Space",
"Relative Complement of Relative Complement",
"Definition:Closed Set/Metric Space",
"Definition:Closed Set/Metric Space",
"Category:Real Intervals",
"Category:Real Number Line with Euclidean Metric"
] |
proofwiki-6275 | Open Real Interval is not Closed Set | Let $\R$ be the real number line with the usual (Euclidean) metric.
Let $I = \openint a b$ be an open real interval.
Then $I$ is not a closed set of $\R$. | Consider the relative complement of $I$ in $\R$:
:$J = \relcomp \R I = \R \setminus I = \hointl \gets a \cup \hointr b \to$
Let $\epsilon \in \R_{>0}$.
Consider the open $\epsilon$-ball $\map {B_\epsilon} a$.
Whatever the value of $\epsilon$ is, $a + \epsilon$ is not in $\map {B_\epsilon} a$.
So, by definition, $J$ is ... | Let $\R$ be the [[Definition:Real Number Line with Euclidean Metric|real number line with the usual (Euclidean) metric]].
Let $I = \openint a b$ be an [[Definition:Open Real Interval|open real interval]].
Then $I$ is not a [[Definition:Closed Set (Metric Space)|closed set]] of $\R$. | Consider the [[Definition:Relative Complement|relative complement]] of $I$ in $\R$:
:$J = \relcomp \R I = \R \setminus I = \hointl \gets a \cup \hointr b \to$
Let $\epsilon \in \R_{>0}$.
Consider the [[Definition:Open Ball|open $\epsilon$-ball]] $\map {B_\epsilon} a$.
Whatever the value of $\epsilon$ is, $a + \epsil... | Open Real Interval is not Closed Set | https://proofwiki.org/wiki/Open_Real_Interval_is_not_Closed_Set | https://proofwiki.org/wiki/Open_Real_Interval_is_not_Closed_Set | [
"Real Intervals",
"Real Number Line with Euclidean Metric"
] | [
"Definition:Euclidean Metric/Real Number Line",
"Definition:Real Interval/Open",
"Definition:Closed Set/Metric Space"
] | [
"Definition:Relative Complement",
"Definition:Open Ball",
"Definition:Open Set/Metric Space",
"Relative Complement of Relative Complement",
"Definition:Closed Set/Metric Space",
"Category:Real Intervals",
"Category:Real Number Line with Euclidean Metric"
] |
proofwiki-6276 | Open Real Interval is not Compact | Let $\R$ be the real number line considered as an Euclidean space.
Let $I = \openint a b$ be an open real interval.
Then $I$ is not compact. | It suffices to demonstrate this for a particular open interval: we use $\openint 0 1$.
Consider the set of open intervals $\openint {\dfrac 1 n} 1$ for all $n \in \Z_{>1}$.
Each of these is an open set in $\openint 0 1$.
Also:
:$\openint 0 1 = \ds \bigcup_{n \mathop \ge 2} \openint {\dfrac 1 n} 1$
Thus $\ds \bigcup_{n ... | Let $\R$ be the [[Definition:Real Number Line|real number line]] considered as an [[Definition:Euclidean Space|Euclidean space]].
Let $I = \openint a b$ be an [[Definition:Open Real Interval|open real interval]].
Then $I$ is not [[Definition:Compact (Real Analysis)|compact]]. | It suffices to demonstrate this for a particular [[Definition:Open Real Interval|open interval]]: we use $\openint 0 1$.
Consider the [[Definition:Set|set]] of [[Definition:Open Real Interval|open intervals]] $\openint {\dfrac 1 n} 1$ for all $n \in \Z_{>1}$.
Each of these is an [[Definition:Open Set (Topology)|open ... | Open Real Interval is not Compact/Proof 2 | https://proofwiki.org/wiki/Open_Real_Interval_is_not_Compact | https://proofwiki.org/wiki/Open_Real_Interval_is_not_Compact/Proof_2 | [
"Open Real Interval is not Compact",
"Real Intervals",
"Examples of Compact Topological Spaces"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space",
"Definition:Real Interval/Open",
"Definition:Compact Space/Real Analysis"
] | [
"Definition:Real Interval/Open",
"Definition:Set",
"Definition:Real Interval/Open",
"Definition:Open Set/Topology",
"Definition:Open Cover",
"Definition:Subcover/Finite",
"Definition:Subcover/Finite",
"Definition:Cover of Set"
] |
proofwiki-6277 | Set of Integers is not Compact | Let $\Z$ be the set of integers.
Then $\Z$ is not compact. | Let $\R$ be the real number line considered as an Euclidean space.
From Set of Integers is not Bounded, $\Z$ is not bounded in $\R$.
The result follows by definition of compact.
{{qed}} | Let $\Z$ be the [[Definition:Integer|set of integers]].
Then $\Z$ is not [[Definition:Compact (Real Analysis)|compact]]. | Let $\R$ be the [[Definition:Real Number Line|real number line]] considered as an [[Definition:Euclidean Space|Euclidean space]].
From [[Set of Integers is not Bounded]], $\Z$ is not [[Definition:Bounded Space|bounded]] in $\R$.
The result follows by definition of [[Definition:Compact (Real Analysis)|compact]].
{{qed... | Set of Integers is not Compact | https://proofwiki.org/wiki/Set_of_Integers_is_not_Compact | https://proofwiki.org/wiki/Set_of_Integers_is_not_Compact | [
"Integers",
"Compact Spaces (Real Analysis)"
] | [
"Definition:Integer",
"Definition:Compact Space/Real Analysis"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space",
"Set of Integers is not Bounded",
"Definition:Bounded Metric Space",
"Definition:Compact Space/Real Analysis"
] |
proofwiki-6278 | Heine-Borel Theorem/Euclidean Space | Let $n \in \N_{> 0}$.
Let $C$ be a subspace of the Euclidean space $\R^n$.
Then $C$ is closed and bounded {{iff}} it is compact. | === Necessary Condition ===
{{:Heine-Borel Theorem/Euclidean Space/Necessary Condition}}{{qed|lemma}} | Let $n \in \N_{> 0}$.
Let $C$ be a [[Definition:Metric Subspace|subspace]] of the [[Definition:Euclidean Space|Euclidean space]] $\R^n$.
Then $C$ is [[Definition:Closed Set (Topology)|closed]] and [[Definition:Bounded Metric Space|bounded]] {{iff}} it is [[Definition:Compact Topological Space|compact]]. | === [[Heine-Borel Theorem/Euclidean Space/Necessary Condition|Necessary Condition]] ===
{{:Heine-Borel Theorem/Euclidean Space/Necessary Condition}}{{qed|lemma}} | Heine-Borel Theorem/Euclidean Space | https://proofwiki.org/wiki/Heine-Borel_Theorem/Euclidean_Space | https://proofwiki.org/wiki/Heine-Borel_Theorem/Euclidean_Space | [
"Heine-Borel Theorem",
"Real Euclidean Spaces"
] | [
"Definition:Metric Subspace",
"Definition:Euclidean Space",
"Definition:Closed Set/Topology",
"Definition:Bounded Metric Space",
"Definition:Compact Topological Space"
] | [
"Heine-Borel Theorem/Euclidean Space/Necessary Condition"
] |
proofwiki-6279 | Heine-Borel Theorem/Euclidean Space | Let $n \in \N_{> 0}$.
Let $C$ be a subspace of the Euclidean space $\R^n$.
Then $C$ is closed and bounded {{iff}} it is compact. | Let $C \subseteq \R^n$ be closed and bounded.
Since $C$ is bounded, $C \subseteq \closedint a b^n = B$ for some $a, b \in \R$.
By the Heine-Borel Theorem: Real Line and by Topological Product of Compact Spaces, it follows that $B$ is compact.
From Euclidean Topology is Product Topology, it follows that $B$ is compact i... | Let $n \in \N_{> 0}$.
Let $C$ be a [[Definition:Metric Subspace|subspace]] of the [[Definition:Euclidean Space|Euclidean space]] $\R^n$.
Then $C$ is [[Definition:Closed Set (Topology)|closed]] and [[Definition:Bounded Metric Space|bounded]] {{iff}} it is [[Definition:Compact Topological Space|compact]]. | Let $C \subseteq \R^n$ be [[Definition:Closed Set (Topology)|closed]] and [[Definition:Bounded Metric Space|bounded]].
Since $C$ is [[Definition:Bounded Metric Space|bounded]], $C \subseteq \closedint a b^n = B$ for some $a, b \in \R$.
By the [[Heine-Borel Theorem/Real Line|Heine-Borel Theorem: Real Line]] and by [[T... | Heine-Borel Theorem/Euclidean Space/Necessary Condition/Proof 1 | https://proofwiki.org/wiki/Heine-Borel_Theorem/Euclidean_Space | https://proofwiki.org/wiki/Heine-Borel_Theorem/Euclidean_Space/Necessary_Condition/Proof_1 | [
"Heine-Borel Theorem",
"Real Euclidean Spaces"
] | [
"Definition:Metric Subspace",
"Definition:Euclidean Space",
"Definition:Closed Set/Topology",
"Definition:Bounded Metric Space",
"Definition:Compact Topological Space"
] | [
"Definition:Closed Set/Topology",
"Definition:Bounded Metric Space",
"Definition:Bounded Metric Space",
"Heine-Borel Theorem/Real Line",
"Topological Product of Compact Spaces",
"Definition:Compact Space/Euclidean Space",
"Euclidean Topology is Product Topology",
"Definition:Compact Space/Euclidean Sp... |
proofwiki-6280 | Heine-Borel Theorem/Euclidean Space | Let $n \in \N_{> 0}$.
Let $C$ be a subspace of the Euclidean space $\R^n$.
Then $C$ is closed and bounded {{iff}} it is compact. | The proof holds for $n = 1$, as follows.
Suppose $C$ is a closed, bounded subspace of $\R$.
Then $C \subseteq \closedint a b$ for some $a, b \in \R$.
Moreover, $C$ is closed in $\closedint a b$ by {{Corollary|Closed Set in Topological Subspace}}.
Hence $C$ is compact, by Closed Subspace of Compact Space is Compact.
Now... | Let $n \in \N_{> 0}$.
Let $C$ be a [[Definition:Metric Subspace|subspace]] of the [[Definition:Euclidean Space|Euclidean space]] $\R^n$.
Then $C$ is [[Definition:Closed Set (Topology)|closed]] and [[Definition:Bounded Metric Space|bounded]] {{iff}} it is [[Definition:Compact Topological Space|compact]]. | The proof holds for $n = 1$, as follows.
Suppose $C$ is a [[Definition:Closed Set (Topology)|closed]], [[Definition:Bounded Metric Space|bounded]] [[Definition:Topological Subspace|subspace]] of $\R$.
Then $C \subseteq \closedint a b$ for some $a, b \in \R$.
Moreover, $C$ is [[Definition:Closed Interval|closed]] in ... | Heine-Borel Theorem/Euclidean Space/Necessary Condition/Proof 2 | https://proofwiki.org/wiki/Heine-Borel_Theorem/Euclidean_Space | https://proofwiki.org/wiki/Heine-Borel_Theorem/Euclidean_Space/Necessary_Condition/Proof_2 | [
"Heine-Borel Theorem",
"Real Euclidean Spaces"
] | [
"Definition:Metric Subspace",
"Definition:Euclidean Space",
"Definition:Closed Set/Topology",
"Definition:Bounded Metric Space",
"Definition:Compact Topological Space"
] | [
"Definition:Closed Set/Topology",
"Definition:Bounded Metric Space",
"Definition:Topological Subspace",
"Definition:Interval/Ordered Set/Closed",
"Definition:Compact Topological Space/Subspace",
"Closed Subspace of Compact Space is Compact",
"Definition:Interval/Ordered Set/Closed",
"Definition:Bounde... |
proofwiki-6281 | Continuous Function on Compact Space is Bounded | Let $\struct {X, \tau}$ be a compact topological space.
Let $\struct {Y, \norm {\, \cdot \, } }$ be a normed vector space.
Let $f: X \to Y$ be a continuous mapping.
Then $f$ is bounded. | {{AimForCont}} $f$ is not bounded.
Let $g : X \to \R$ be defined by:
:$\map g x = \norm {\map f x}$
From Norm is Continuous and Composite of Continuous Mappings is Continuous, it follows that $g$ is continuous.
For all $n \in \N$, set $A_n := g^{-1} \sqbrk {\map {B_n} 0}$, where $\map {B_n} 0$ denotes the open ball in ... | Let $\struct {X, \tau}$ be a [[Definition:Compact Topological Space|compact topological space]].
Let $\struct {Y, \norm {\, \cdot \, } }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $f: X \to Y$ be a [[Definition:Continuous Mapping (Topology)|continuous mapping]].
Then $f$ is [[Definition:Bound... | {{AimForCont}} $f$ is not [[Definition:Bounded Mapping|bounded]].
Let $g : X \to \R$ be defined by:
:$\map g x = \norm {\map f x}$
From [[Norm is Continuous]] and [[Composite of Continuous Mappings is Continuous]], it follows that $g$ is [[Definition:Continuous Mapping (Topology)|continuous]].
For all $n \in \N$, se... | Continuous Function on Compact Space is Bounded/Proof 1 | https://proofwiki.org/wiki/Continuous_Function_on_Compact_Space_is_Bounded | https://proofwiki.org/wiki/Continuous_Function_on_Compact_Space_is_Bounded/Proof_1 | [
"Continuous Function on Compact Space is Bounded",
"Continuous Functions",
"Compact Topological Spaces",
"Normed Vector Spaces"
] | [
"Definition:Compact Topological Space",
"Definition:Normed Vector Space",
"Definition:Continuous Mapping (Topology)",
"Definition:Bounded Mapping"
] | [
"Definition:Bounded Mapping",
"Norm is Continuous",
"Composite of Continuous Mappings is Continuous",
"Definition:Continuous Mapping (Topology)",
"Definition:Open Ball/Real Analysis",
"Definition:Open Ball/Radius",
"Definition:Open Ball/Center",
"Open Ball is Open Set/Normed Vector Space",
"Definiti... |
proofwiki-6282 | Continuous Function on Compact Space is Bounded | Let $\struct {X, \tau}$ be a compact topological space.
Let $\struct {Y, \norm {\, \cdot \, } }$ be a normed vector space.
Let $f: X \to Y$ be a continuous mapping.
Then $f$ is bounded. | From Continuous Image of Compact Space is Compact, $f \sqbrk X$ is a compact subset of $Y$.
From Compact Subset of Normed Vector Space is Closed and Bounded, $f \sqbrk X$ is bounded.
Hence there exists a real number $M > 0$ such that:
:$\norm {\map f x} \le M$ for all $x \in X$.
So $f$ is bounded.
{{qed}} | Let $\struct {X, \tau}$ be a [[Definition:Compact Topological Space|compact topological space]].
Let $\struct {Y, \norm {\, \cdot \, } }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $f: X \to Y$ be a [[Definition:Continuous Mapping (Topology)|continuous mapping]].
Then $f$ is [[Definition:Bound... | From [[Continuous Image of Compact Space is Compact]], $f \sqbrk X$ is a [[Definition:Compact Topological Space|compact subset]] of $Y$.
From [[Compact Subset of Normed Vector Space is Closed and Bounded]], $f \sqbrk X$ is [[Definition:Bounded Subset of Normed Vector Space|bounded]].
Hence there exists a [[Definitio... | Continuous Function on Compact Space is Bounded/Proof 2 | https://proofwiki.org/wiki/Continuous_Function_on_Compact_Space_is_Bounded | https://proofwiki.org/wiki/Continuous_Function_on_Compact_Space_is_Bounded/Proof_2 | [
"Continuous Function on Compact Space is Bounded",
"Continuous Functions",
"Compact Topological Spaces",
"Normed Vector Spaces"
] | [
"Definition:Compact Topological Space",
"Definition:Normed Vector Space",
"Definition:Continuous Mapping (Topology)",
"Definition:Bounded Mapping"
] | [
"Continuous Image of Compact Space is Compact",
"Definition:Compact Topological Space",
"Compact Subset of Normed Vector Space is Closed and Bounded",
"Definition:Bounded Subset of Normed Vector Space",
"Definition:Real Number",
"Definition:Bounded Mapping"
] |
proofwiki-6283 | Continuous Function on Compact Space is Uniformly Continuous | Let $\R^n$ be the $n$-dimensional Euclidean space.
Let $S \subseteq \R^n$ be a compact subspace of $\R^n$.
Let $f: S \to \R$ be a continuous function.
Then $f$ is uniformly continuous on $S$. | {{ProofWanted|Use Heine-Cantor Theorem}} | Let $\R^n$ be the [[Definition:Euclidean Space|$n$-dimensional Euclidean space]].
Let $S \subseteq \R^n$ be a [[Definition:Compact (Real Analysis)|compact subspace]] of $\R^n$.
Let $f: S \to \R$ be a [[Definition:Continuous Mapping (Metric Spaces)|continuous function]].
Then $f$ is [[Definition:Uniformly Continuous... | {{ProofWanted|Use [[Heine-Cantor Theorem]]}} | Continuous Function on Compact Space is Uniformly Continuous | https://proofwiki.org/wiki/Continuous_Function_on_Compact_Space_is_Uniformly_Continuous | https://proofwiki.org/wiki/Continuous_Function_on_Compact_Space_is_Uniformly_Continuous | [
"Continuous Functions",
"Uniformly Continuous Functions"
] | [
"Definition:Euclidean Space",
"Definition:Compact Space/Real Analysis",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Uniform Continuity/Metric Space"
] | [
"Heine-Cantor Theorem"
] |
proofwiki-6284 | T2 Space is Noetherian iff Finite | Let $\struct {S, \tau}$ be a $T_2$ (Hausdorff) space.
Then $\struct {S, \tau}$ is Noetherian {{iff}} $S$ is finite. | === Necessary Condition ===
Let $\struct {S, \tau}$ be Noetherian.
{{Recall|Noetherian Topological Space|Noetherian topological space|index = 5}}
{{:Definition:Noetherian Topological Space/Definition 5}}
Let $H \subseteq S$ be a subspace of $\struct {S, \tau}$.
From Compact Subspace of Hausdorff Space is Closed, $H$ is... | Let $\struct {S, \tau}$ be a [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
Then $\struct {S, \tau}$ is [[Definition:Noetherian Topological Space|Noetherian]] {{iff}} $S$ is [[Definition:Finite Set|finite]]. | === Necessary Condition ===
Let $\struct {S, \tau}$ be [[Definition:Noetherian Topological Space|Noetherian]].
{{Recall|Noetherian Topological Space|Noetherian topological space|index = 5}}
{{:Definition:Noetherian Topological Space/Definition 5}}
Let $H \subseteq S$ be a [[Definition:Subspace|subspace]] of $\struct... | T2 Space is Noetherian iff Finite | https://proofwiki.org/wiki/T2_Space_is_Noetherian_iff_Finite | https://proofwiki.org/wiki/T2_Space_is_Noetherian_iff_Finite | [
"Hausdorff Spaces",
"Finite Topological Spaces",
"Noetherian Topological Spaces"
] | [
"Definition:T2 Space",
"Definition:Noetherian Topological Space",
"Definition:Finite Set"
] | [
"Definition:Noetherian Topological Space",
"Definition:Subspace",
"Compact Subspace of Hausdorff Space is Closed",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Discrete Topology",
"Definition:Discrete Topology",
"Definition:Compact Topological Space",
"Discrete Spa... |
proofwiki-6285 | Indiscrete Space is Noetherian | Let $\struct {S, \tau}$ be an indiscrete topological space.
Then $\struct {S, \tau}$ is Noetherian. | Let $\struct {H, \tau_H}$ be a subspace of $T$.
From Subset of Indiscrete Space is Compact, $\struct {H, \tau_H}$ is compact.
The result follows by definition of Noetherian topological space.
{{qed}} | Let $\struct {S, \tau}$ be an [[Definition:Indiscrete Space|indiscrete topological space]].
Then $\struct {S, \tau}$ is [[Definition:Noetherian Topological Space|Noetherian]]. | Let $\struct {H, \tau_H}$ be a [[Definition:Topological Subspace|subspace]] of $T$.
From [[Subset of Indiscrete Space is Compact]], $\struct {H, \tau_H}$ is [[Definition:Compact Topological Space|compact]].
The result follows by definition of [[Definition:Noetherian Topological Space|Noetherian topological space]].
{... | Indiscrete Space is Noetherian | https://proofwiki.org/wiki/Indiscrete_Space_is_Noetherian | https://proofwiki.org/wiki/Indiscrete_Space_is_Noetherian | [
"Indiscrete Topology",
"Examples of Noetherian Topological Spaces"
] | [
"Definition:Indiscrete Topology",
"Definition:Noetherian Topological Space"
] | [
"Definition:Topological Subspace",
"Subset of Indiscrete Space is Compact",
"Definition:Compact Topological Space",
"Definition:Noetherian Topological Space"
] |
proofwiki-6286 | Complete and Totally Bounded Metric Space is Sequentially Compact | Let $M = \struct {A, d}$ be a metric space.
Let $M$ be complete and totally bounded.
Then $M$ is sequentially compact. | Let $\sequence {x_m}_{m \mathop \in \N}$ be an infinite sequence in $A$.
By the definition of a totally bounded metric space, we can use the {{Axiom-link|Countable Choice}} to obtain a sequence $\sequence {F_n}_{n \mathop \in \N}$ such that:
:For all $n \in \N$, $F_n$ is a finite $2^{-n}$-net for $M$.
For all $n \in \N... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $M$ be [[Definition:Complete Metric Space|complete]] and [[Definition:Totally Bounded Metric Space|totally bounded]].
Then $M$ is [[Definition:Sequentially Compact Space|sequentially compact]]. | Let $\sequence {x_m}_{m \mathop \in \N}$ be an [[Definition:Infinite Sequence|infinite sequence]] in $A$.
By the definition of a [[Definition:Totally Bounded Metric Space|totally bounded]] [[Definition:Metric Space|metric space]], we can use the {{Axiom-link|Countable Choice}} to obtain a [[Definition:Sequence|sequen... | Complete and Totally Bounded Metric Space is Sequentially Compact/Proof 1 | https://proofwiki.org/wiki/Complete_and_Totally_Bounded_Metric_Space_is_Sequentially_Compact | https://proofwiki.org/wiki/Complete_and_Totally_Bounded_Metric_Space_is_Sequentially_Compact/Proof_1 | [
"Complete Metric Spaces",
"Sequentially Compact Spaces",
"Totally Bounded Metric Spaces",
"Complete and Totally Bounded Metric Space is Sequentially Compact"
] | [
"Definition:Metric Space",
"Definition:Complete Metric Space",
"Definition:Totally Bounded Metric Space",
"Definition:Sequentially Compact Space"
] | [
"Definition:Sequence/Infinite Sequence",
"Definition:Totally Bounded Metric Space",
"Definition:Metric Space",
"Definition:Sequence",
"Definition:Epsilon-Net/Finite Net",
"Definition:Epsilon-Net",
"Definition:Infinite Set",
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Definition:Infinite ... |
proofwiki-6287 | Complete and Totally Bounded Metric Space is Sequentially Compact | Let $M = \struct {A, d}$ be a metric space.
Let $M$ be complete and totally bounded.
Then $M$ is sequentially compact. | The results:
:Compact Space is Countably Compact
:Countably Compact Metric Space is Sequentially Compact
show that it suffices to prove that $M$ is compact.
{{AimForCont}} that $M$ is not compact.
Let $\CC$ be an open cover for $A$ such that $\CC$ does not have a finite subcover for $A$.
By the definition of a totally ... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $M$ be [[Definition:Complete Metric Space|complete]] and [[Definition:Totally Bounded Metric Space|totally bounded]].
Then $M$ is [[Definition:Sequentially Compact Space|sequentially compact]]. | The results:
:[[Compact Space is Countably Compact]]
:[[Countably Compact Metric Space is Sequentially Compact]]
show that it suffices to prove that $M$ is [[Definition:Compact Metric Space|compact]].
{{AimForCont}} that $M$ is not [[Definition:Compact Metric Space|compact]].
Let $\CC$ be an [[Definition:Open Cover|... | Complete and Totally Bounded Metric Space is Sequentially Compact/Proof 2 | https://proofwiki.org/wiki/Complete_and_Totally_Bounded_Metric_Space_is_Sequentially_Compact | https://proofwiki.org/wiki/Complete_and_Totally_Bounded_Metric_Space_is_Sequentially_Compact/Proof_2 | [
"Complete Metric Spaces",
"Sequentially Compact Spaces",
"Totally Bounded Metric Spaces",
"Complete and Totally Bounded Metric Space is Sequentially Compact"
] | [
"Definition:Metric Space",
"Definition:Complete Metric Space",
"Definition:Totally Bounded Metric Space",
"Definition:Sequentially Compact Space"
] | [
"Compact Space is Countably Compact",
"Countably Compact Metric Space is Sequentially Compact",
"Definition:Compact Space/Metric Space",
"Definition:Compact Space/Metric Space",
"Definition:Open Cover",
"Definition:Subcover/Finite",
"Definition:Totally Bounded Metric Space",
"Definition:Metric Space",... |
proofwiki-6288 | Complete and Totally Bounded Metric Space is Sequentially Compact | Let $M = \struct {A, d}$ be a metric space.
Let $M$ be complete and totally bounded.
Then $M$ is sequentially compact. | Let $M$ be both complete and totally bounded.
Let $\sequence {a_k}$ be any infinite sequence in $A$.
Let $\epsilon \in \R_{>0}$.
Let $x_1, \ldots, x_n \in X$ be a finite set of points such that:
:$\ds A = \bigcup_{i \mathop = 1}^n \map {B_\epsilon} {x_i}$
where $\map {B_\epsilon} {x_i}$ represents the open $\epsilon$-b... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $M$ be [[Definition:Complete Metric Space|complete]] and [[Definition:Totally Bounded Metric Space|totally bounded]].
Then $M$ is [[Definition:Sequentially Compact Space|sequentially compact]]. | Let $M$ be both [[Definition:Complete Metric Space|complete]] and [[Definition:Totally Bounded Metric Space|totally bounded]].
Let $\sequence {a_k}$ be any [[Definition:Infinite Sequence|infinite sequence]] in $A$.
Let $\epsilon \in \R_{>0}$.
Let $x_1, \ldots, x_n \in X$ be a [[Definition:Finite Set|finite set]] of ... | Complete and Totally Bounded Metric Space is Sequentially Compact/Proof 3 | https://proofwiki.org/wiki/Complete_and_Totally_Bounded_Metric_Space_is_Sequentially_Compact | https://proofwiki.org/wiki/Complete_and_Totally_Bounded_Metric_Space_is_Sequentially_Compact/Proof_3 | [
"Complete Metric Spaces",
"Sequentially Compact Spaces",
"Totally Bounded Metric Spaces",
"Complete and Totally Bounded Metric Space is Sequentially Compact"
] | [
"Definition:Metric Space",
"Definition:Complete Metric Space",
"Definition:Totally Bounded Metric Space",
"Definition:Sequentially Compact Space"
] | [
"Definition:Complete Metric Space",
"Definition:Totally Bounded Metric Space",
"Definition:Sequence/Infinite Sequence",
"Definition:Finite Set",
"Definition:Open Ball",
"Definition:Totally Bounded Metric Space",
"Definition:Sequence/Infinite Sequence",
"Axiom:Axiom of Countable Choice",
"Axiom:Axiom... |
proofwiki-6289 | Complete and Totally Bounded Metric Space is Sequentially Compact | Let $M = \struct {A, d}$ be a metric space.
Let $M$ be complete and totally bounded.
Then $M$ is sequentially compact. | We use the following {{Lemma|Complete and Totally Bounded Metric Space is Sequentially Compact}}, which depends on the {{Axiom-link|Countable Choice}}.
=== Lemma ===
{{:Complete and Totally Bounded Metric Space is Sequentially Compact/Lemma}}{{qed|lemma}}
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $A$.
I... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $M$ be [[Definition:Complete Metric Space|complete]] and [[Definition:Totally Bounded Metric Space|totally bounded]].
Then $M$ is [[Definition:Sequentially Compact Space|sequentially compact]]. | We use the following {{Lemma|Complete and Totally Bounded Metric Space is Sequentially Compact}}, which depends on the {{Axiom-link|Countable Choice}}.
=== [[Complete and Totally Bounded Metric Space is Sequentially Compact/Lemma|Lemma]] ===
{{:Complete and Totally Bounded Metric Space is Sequentially Compact/Lemma}}{... | Complete and Totally Bounded Metric Space is Sequentially Compact/Proof 4 | https://proofwiki.org/wiki/Complete_and_Totally_Bounded_Metric_Space_is_Sequentially_Compact | https://proofwiki.org/wiki/Complete_and_Totally_Bounded_Metric_Space_is_Sequentially_Compact/Proof_4 | [
"Complete Metric Spaces",
"Sequentially Compact Spaces",
"Totally Bounded Metric Spaces",
"Complete and Totally Bounded Metric Space is Sequentially Compact"
] | [
"Definition:Metric Space",
"Definition:Complete Metric Space",
"Definition:Totally Bounded Metric Space",
"Definition:Sequentially Compact Space"
] | [
"Complete and Totally Bounded Metric Space is Sequentially Compact/Lemma",
"Definition:Sequence/Infinite Sequence",
"Definition:Recursively Defined Mapping/Natural Numbers",
"Definition:Subsequence",
"Complete and Totally Bounded Metric Space is Sequentially Compact/Lemma",
"Definition:Subsequence",
"De... |
proofwiki-6290 | Closed Bounded Subset of Real Numbers is Compact | Let $\R$ be the real number line considered as an Euclidean space.
Let $S \subseteq \R$ be a closed and bounded subspace of $\R$.
Then $S$ is compact in $\R$. | A closed and bounded subspace $S$ of $\R$ is a closed subspace of some closed real interval $\closedint a b$.
From Closed Subspace of Compact Space is Compact, it suffices to prove that $\closedint a b$ is compact.
Let $\UU$ be any open cover of $\closedint a b$.
Let:
:$G = \set {x \in \R: x \ge a, \closedint a x \text... | Let $\R$ be the [[Definition:Real Number Line|real number line]] considered as an [[Definition:Euclidean Space|Euclidean space]].
Let $S \subseteq \R$ be a [[Definition:Closed Set (Metric Space)|closed]] and [[Definition:Bounded Metric Space|bounded]] [[Definition:Metric Subspace|subspace]] of [[Definition:Real Number... | A [[Definition:Closed Set (Metric Space)|closed]] and [[Definition:Bounded Metric Space|bounded]] [[Definition:Metric Subspace|subspace]] $S$ of $\R$ is a [[Definition:Closed Set (Metric Space)|closed]] [[Definition:Metric Subspace|subspace]] of some [[Definition:Closed Real Interval|closed real interval]] $\closedint ... | Closed Bounded Subset of Real Numbers is Compact/Proof 1 | https://proofwiki.org/wiki/Closed_Bounded_Subset_of_Real_Numbers_is_Compact | https://proofwiki.org/wiki/Closed_Bounded_Subset_of_Real_Numbers_is_Compact/Proof_1 | [
"Closed Bounded Subset of Real Numbers is Compact",
"Euclidean Spaces",
"Real Analysis"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space",
"Definition:Closed Set/Metric Space",
"Definition:Bounded Metric Space",
"Definition:Metric Subspace",
"Definition:Real Number",
"Definition:Compact Topological Space/Subspace"
] | [
"Definition:Closed Set/Metric Space",
"Definition:Bounded Metric Space",
"Definition:Metric Subspace",
"Definition:Closed Set/Metric Space",
"Definition:Metric Subspace",
"Definition:Real Interval/Closed",
"Closed Subspace of Compact Space is Compact",
"Definition:Compact Space/Real Analysis",
"Defi... |
proofwiki-6291 | Closed Bounded Subset of Real Numbers is Compact | Let $\R$ be the real number line considered as an Euclidean space.
Let $S \subseteq \R$ be a closed and bounded subspace of $\R$.
Then $S$ is compact in $\R$. | Let $S$ be closed and bounded.
As $S$ is bounded, there exist some $a, b \in \R$ such that:
:$S \subseteq \openint a b$
where $\openint a b$ is the open interval between $a$ and $b$.
It follows that $S \subseteq \closedint a b$.
Consider the set:
:$U = \relcomp \R S \cap \openint {a - 1} {b + 1}$
By inspection it can b... | Let $\R$ be the [[Definition:Real Number Line|real number line]] considered as an [[Definition:Euclidean Space|Euclidean space]].
Let $S \subseteq \R$ be a [[Definition:Closed Set (Metric Space)|closed]] and [[Definition:Bounded Metric Space|bounded]] [[Definition:Metric Subspace|subspace]] of [[Definition:Real Number... | Let $S$ be [[Definition:Closed Set (Metric Space)|closed]] and [[Definition:Bounded Metric Space|bounded]].
As $S$ is [[Definition:Bounded Metric Space|bounded]], there exist some $a, b \in \R$ such that:
:$S \subseteq \openint a b$
where $\openint a b$ is the [[Definition:Open Real Interval|open interval]] between $a... | Closed Bounded Subset of Real Numbers is Compact/Proof 2 | https://proofwiki.org/wiki/Closed_Bounded_Subset_of_Real_Numbers_is_Compact | https://proofwiki.org/wiki/Closed_Bounded_Subset_of_Real_Numbers_is_Compact/Proof_2 | [
"Closed Bounded Subset of Real Numbers is Compact",
"Euclidean Spaces",
"Real Analysis"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space",
"Definition:Closed Set/Metric Space",
"Definition:Bounded Metric Space",
"Definition:Metric Subspace",
"Definition:Real Number",
"Definition:Compact Topological Space/Subspace"
] | [
"Definition:Closed Set/Metric Space",
"Definition:Bounded Metric Space",
"Definition:Bounded Metric Space",
"Definition:Real Interval/Open",
"Union of Open Sets of Metric Space is Open",
"Definition:Open Set/Metric Space",
"Definition:Open Cover",
"Definition:Open Cover",
"Definition:Subcover/Finite... |
proofwiki-6292 | Non-Closed Set of Real Numbers is not Compact | Let $\R$ be the set of real numbers considered as an Euclidean space.
Let $S \subseteq \R$ be non-closed in $\R$.
Then $S$ is not a compact subspace of $\R$. | Consider the complement of $S$ in $\R$:
:$S' = \relcomp \R S = \R \setminus S$
As $S$ is not closed, by definition $S'$ is not open.
Thus by definition there exists $x \in S'$ such that:
:$\forall \epsilon \in \R_{>0}: \map {B_\epsilon} x \notin S'$
where $\map {B_\epsilon} x$ denotes the open $\epsilon$-ball of $x$.
T... | Let $\R$ be the [[Definition:Real Number|set of real numbers]] considered as an [[Definition:Euclidean Space|Euclidean space]].
Let $S \subseteq \R$ be non-[[Definition:Closed Set (Metric Space)|closed]] in $\R$.
Then $S$ is not a [[Definition:Compact Subspace|compact subspace]] of $\R$. | Consider the [[Definition:Relative Complement|complement of $S$ in $\R$]]:
:$S' = \relcomp \R S = \R \setminus S$
As $S$ is not [[Definition:Closed Set (Metric Space)|closed]], by definition $S'$ is not [[Definition:Open Set (Metric Space)|open]].
Thus by definition there exists $x \in S'$ such that:
:$\forall \epsi... | Non-Closed Set of Real Numbers is not Compact/Proof 1 | https://proofwiki.org/wiki/Non-Closed_Set_of_Real_Numbers_is_not_Compact | https://proofwiki.org/wiki/Non-Closed_Set_of_Real_Numbers_is_not_Compact/Proof_1 | [
"Non-Closed Set of Real Numbers is not Compact",
"Compact Spaces (Real Analysis)",
"Real Analysis"
] | [
"Definition:Real Number",
"Definition:Euclidean Space",
"Definition:Closed Set/Metric Space",
"Definition:Compact Topological Space/Subspace"
] | [
"Definition:Relative Complement",
"Definition:Closed Set/Metric Space",
"Definition:Open Set/Metric Space",
"Definition:Open Ball",
"Union of Open Sets of Metric Space is Open",
"Definition:Open Set/Metric Space",
"Definition:Open Cover",
"Definition:Finite Set",
"Definition:Open Cover",
"Definiti... |
proofwiki-6293 | Non-Closed Set of Real Numbers is not Compact | Let $\R$ be the set of real numbers considered as an Euclidean space.
Let $S \subseteq \R$ be non-closed in $\R$.
Then $S$ is not a compact subspace of $\R$. | From:
:Real Number Line is Metric Space
:Metric Space is Hausdorff
:Compact Subspace of Hausdorff Space is Closed
the result follows by the rule of transposition.
{{qed}} | Let $\R$ be the [[Definition:Real Number|set of real numbers]] considered as an [[Definition:Euclidean Space|Euclidean space]].
Let $S \subseteq \R$ be non-[[Definition:Closed Set (Metric Space)|closed]] in $\R$.
Then $S$ is not a [[Definition:Compact Subspace|compact subspace]] of $\R$. | From:
:[[Real Number Line is Metric Space]]
:[[Metric Space is Hausdorff]]
:[[Compact Subspace of Hausdorff Space is Closed]]
the result follows by the [[Rule of Transposition|rule of transposition]].
{{qed}} | Non-Closed Set of Real Numbers is not Compact/Proof 2 | https://proofwiki.org/wiki/Non-Closed_Set_of_Real_Numbers_is_not_Compact | https://proofwiki.org/wiki/Non-Closed_Set_of_Real_Numbers_is_not_Compact/Proof_2 | [
"Non-Closed Set of Real Numbers is not Compact",
"Compact Spaces (Real Analysis)",
"Real Analysis"
] | [
"Definition:Real Number",
"Definition:Euclidean Space",
"Definition:Closed Set/Metric Space",
"Definition:Compact Topological Space/Subspace"
] | [
"Real Number Line is Metric Space",
"Metric Space is T2",
"Compact Subspace of Hausdorff Space is Closed",
"Rule of Transposition"
] |
proofwiki-6294 | Non-Closed Set of Real Numbers is not Compact | Let $\R$ be the set of real numbers considered as an Euclidean space.
Let $S \subseteq \R$ be non-closed in $\R$.
Then $S$ is not a compact subspace of $\R$. | By the rule of transposition, it suffices to show that if $S$ is a compact subspace of $\R$, then $S$ is closed.
Consider the relative complement of $S$ in $\R$:
:$T = \relcomp \R S = \R \setminus S$
It remains to be shown that $T$ is open in $\R$.
Let $x \in T$.
For all strictly positive real numbers $\epsilon \in \R_... | Let $\R$ be the [[Definition:Real Number|set of real numbers]] considered as an [[Definition:Euclidean Space|Euclidean space]].
Let $S \subseteq \R$ be non-[[Definition:Closed Set (Metric Space)|closed]] in $\R$.
Then $S$ is not a [[Definition:Compact Subspace|compact subspace]] of $\R$. | By the [[Rule of Transposition|rule of transposition]], it suffices to show that if $S$ is a [[Definition:Compact Subspace|compact subspace]] of $\R$, then $S$ is [[Definition:Closed Set (Metric Space)|closed]].
Consider the [[Definition:Relative Complement|relative complement of $S$ in $\R$]]:
:$T = \relcomp \R S = ... | Non-Closed Set of Real Numbers is not Compact/Proof 3 | https://proofwiki.org/wiki/Non-Closed_Set_of_Real_Numbers_is_not_Compact | https://proofwiki.org/wiki/Non-Closed_Set_of_Real_Numbers_is_not_Compact/Proof_3 | [
"Non-Closed Set of Real Numbers is not Compact",
"Compact Spaces (Real Analysis)",
"Real Analysis"
] | [
"Definition:Real Number",
"Definition:Euclidean Space",
"Definition:Closed Set/Metric Space",
"Definition:Compact Topological Space/Subspace"
] | [
"Rule of Transposition",
"Definition:Compact Topological Space/Subspace",
"Definition:Closed Set/Metric Space",
"Definition:Relative Complement",
"Definition:Open Set/Metric Space",
"Definition:Strictly Positive/Real Number",
"Union of Open Sets of Metric Space is Open",
"Definition:Open Set/Metric Sp... |
proofwiki-6295 | Compact Subspace of Real Numbers is Closed and Bounded | Let $\R$ be the real number line considered as a Euclidean space.
Let $S \subseteq \R$ be compact subspace of $\R$.
Then $S$ is closed and bounded in $\R$. | From:
: Non-Closed Set of Real Numbers is not Compact
: Unbounded Set of Real Numbers is not Compact
the result follows by the Rule of Transposition.
{{qed}} | Let $\R$ be the [[Definition:Real Number Line|real number line]] considered as a [[Definition:Euclidean Space|Euclidean space]].
Let $S \subseteq \R$ be [[Definition:Compact Subspace|compact]] [[Definition:Metric Subspace|subspace]] of $\R$.
Then $S$ is [[Definition:Closed Set (Metric Space)|closed]] and [[Definitio... | From:
: [[Non-Closed Set of Real Numbers is not Compact]]
: [[Unbounded Set of Real Numbers is not Compact]]
the result follows by the [[Rule of Transposition]].
{{qed}} | Compact Subspace of Real Numbers is Closed and Bounded/Proof 1 | https://proofwiki.org/wiki/Compact_Subspace_of_Real_Numbers_is_Closed_and_Bounded | https://proofwiki.org/wiki/Compact_Subspace_of_Real_Numbers_is_Closed_and_Bounded/Proof_1 | [
"Compact Subspace of Real Numbers is Closed and Bounded",
"Compact Spaces (Real Analysis)",
"Real Analysis"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space",
"Definition:Compact Topological Space/Subspace",
"Definition:Metric Subspace",
"Definition:Closed Set/Metric Space",
"Definition:Bounded Metric Space"
] | [
"Non-Closed Set of Real Numbers is not Compact",
"Unbounded Set of Real Numbers is not Compact",
"Rule of Transposition"
] |
proofwiki-6296 | Compact Subspace of Real Numbers is Closed and Bounded | Let $\R$ be the real number line considered as a Euclidean space.
Let $S \subseteq \R$ be compact subspace of $\R$.
Then $S$ is closed and bounded in $\R$. | From Real Number Line is Metric Space, $\left({\R, d}\right)$ is a metric space, where $d$ denotes the Euclidean metric on $\R$.
Therefore, the result follows from:
: Metric Space is Hausdorff
: Compact Subspace of Hausdorff Space is Closed
and:
: Compact Metric Space is Totally Bounded
: Totally Bounded Metric Space i... | Let $\R$ be the [[Definition:Real Number Line|real number line]] considered as a [[Definition:Euclidean Space|Euclidean space]].
Let $S \subseteq \R$ be [[Definition:Compact Subspace|compact]] [[Definition:Metric Subspace|subspace]] of $\R$.
Then $S$ is [[Definition:Closed Set (Metric Space)|closed]] and [[Definitio... | From [[Real Number Line is Metric Space]], $\left({\R, d}\right)$ is a [[Definition:Metric Space|metric space]], where $d$ denotes the [[Definition:Euclidean Metric on Real Number Line|Euclidean metric on $\R$]].
Therefore, the result follows from:
: [[Metric Space is Hausdorff]]
: [[Compact Subspace of Hausdorff Spac... | Compact Subspace of Real Numbers is Closed and Bounded/Proof 2 | https://proofwiki.org/wiki/Compact_Subspace_of_Real_Numbers_is_Closed_and_Bounded | https://proofwiki.org/wiki/Compact_Subspace_of_Real_Numbers_is_Closed_and_Bounded/Proof_2 | [
"Compact Subspace of Real Numbers is Closed and Bounded",
"Compact Spaces (Real Analysis)",
"Real Analysis"
] | [
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space",
"Definition:Compact Topological Space/Subspace",
"Definition:Metric Subspace",
"Definition:Closed Set/Metric Space",
"Definition:Bounded Metric Space"
] | [
"Real Number Line is Metric Space",
"Definition:Metric Space",
"Definition:Euclidean Metric/Real Number Line",
"Metric Space is T2",
"Compact Subspace of Hausdorff Space is Closed",
"Compact Metric Space is Totally Bounded",
"Totally Bounded Metric Space is Bounded"
] |
proofwiki-6297 | Category of Subobjects is Category | Let $\mathbf C$ be a metacategory.
Let $C$ be an object of $\mathbf C$.
Let $\map {\mathbf{Sub}_{\mathbf C}} C$ be the category of subobjects of $C$.
Then $\map {\mathbf{Sub}_{\mathbf C}} C$ is a metacategory. | Let us verify the axioms $(C1)$ up to $(C3)$ for a metacategory.
Let $f: m_1 \to m_2$ and $g: m_2 \to m_3$ be morphisms of $\map {\mathbf{Sub}_{\mathbf C}} C$.
That $g \circ f: m_1 \to m_3$ is again a morphism follows from the following commutative diagram in $\mathbf C$:
$\quad\quad \begin{xy}\xymatrix@+2em{
\operator... | Let $\mathbf C$ be a [[Definition:Metacategory|metacategory]].
Let $C$ be an [[Definition:Object (Category Theory)|object]] of $\mathbf C$.
Let $\map {\mathbf{Sub}_{\mathbf C}} C$ be the [[Definition:Category of Subobjects|category of subobjects]] of $C$.
Then $\map {\mathbf{Sub}_{\mathbf C}} C$ is a [[Definition:M... | Let us verify the axioms $(C1)$ up to $(C3)$ for a [[Definition:Metacategory|metacategory]].
Let $f: m_1 \to m_2$ and $g: m_2 \to m_3$ be [[Definition:Morphism (Category Theory)|morphisms]] of $\map {\mathbf{Sub}_{\mathbf C}} C$.
That $g \circ f: m_1 \to m_3$ is again a [[Definition:Morphism (Category Theory)|morphi... | Category of Subobjects is Category | https://proofwiki.org/wiki/Category_of_Subobjects_is_Category | https://proofwiki.org/wiki/Category_of_Subobjects_is_Category | [
"Categories of Subobjects"
] | [
"Definition:Metacategory",
"Definition:Object (Category Theory)",
"Definition:Category of Subobjects",
"Definition:Metacategory"
] | [
"Definition:Metacategory",
"Definition:Morphism",
"Definition:Morphism",
"Definition:Commutative Diagram",
"Definition:Subobject",
"Definition:Identity Morphism",
"Definition:Morphism",
"Definition:Composition of Morphisms",
"Definition:Associative Operation",
"Definition:Metacategory"
] |
proofwiki-6298 | Category of Subobjects is Preorder Category | Let $\mathbf C$ be a metacategory.
Let $C$ be an object of $\mathbf C$.
Let $\map {\mathbf{Sub}_{\mathbf C} } C$ be the category of subobjects of $C$.
Then $\map {\mathbf{Sub}_{\mathbf C} } C$ is a preorder category. | By Category of Subobjects is Category, we know $\map {\mathbf{Sub}_{\mathbf C} } C$ is a metacategory.
By definition of preorder category, it suffices to show that if $f, g: m \to m'$ are morphisms with the same domain and codomain, then $f = g$.
The situation is sketched by the following commutative diagram in $\mathb... | Let $\mathbf C$ be a [[Definition:Metacategory|metacategory]].
Let $C$ be an [[Definition:Object (Category Theory)|object]] of $\mathbf C$.
Let $\map {\mathbf{Sub}_{\mathbf C} } C$ be the [[Definition:Category of Subobjects|category of subobjects]] of $C$.
Then $\map {\mathbf{Sub}_{\mathbf C} } C$ is a [[Definition... | By [[Category of Subobjects is Category]], we know $\map {\mathbf{Sub}_{\mathbf C} } C$ is a [[Definition:Metacategory|metacategory]].
By definition of [[Definition:Preorder Category|preorder category]], it suffices to show that if $f, g: m \to m'$ are [[Definition:Morphism|morphisms]] with the same [[Definition:Domai... | Category of Subobjects is Preorder Category | https://proofwiki.org/wiki/Category_of_Subobjects_is_Preorder_Category | https://proofwiki.org/wiki/Category_of_Subobjects_is_Preorder_Category | [
"Categories of Subobjects",
"Preorder Categories"
] | [
"Definition:Metacategory",
"Definition:Object (Category Theory)",
"Definition:Category of Subobjects",
"Definition:Preorder Category"
] | [
"Category of Subobjects is Category",
"Definition:Metacategory",
"Definition:Preorder Category",
"Definition:Morphism",
"Definition:Domain (Category Theory)",
"Definition:Codomain (Category Theory)",
"Definition:Commutative Diagram",
"Definition:Subobject",
"Definition:Monomorphism (Category Theory)... |
proofwiki-6299 | Integral of Power/Fermat's Proof | :$\ds \forall n \in \Q_{>0}: \int_0^b x^n \rd x = \frac {b^{n + 1} } {n + 1}$ | First let $n$ be a positive integer.
Take a real number $r \in \R$ such that $0 < r < 1$ but reasonably close to $1$.
Consider a subdivision $S$ of the closed interval $\closedint 0 b$ defined as:
:$S = \set {0, \ldots, r^2 b, r b, b}$
that is, by taking as the points of subdivision successive powers of $r$.
Now we tak... | :$\ds \forall n \in \Q_{>0}: \int_0^b x^n \rd x = \frac {b^{n + 1} } {n + 1}$ | First let $n$ be a [[Definition:Positive Integer|positive integer]].
Take a [[Definition:Real Number|real number]] $r \in \R$ such that $0 < r < 1$ but reasonably close to $1$.
Consider a [[Definition:Subdivision of Interval|subdivision]] $S$ of the [[Definition:Closed Real Interval|closed interval]] $\closedint 0 b$... | Integral of Power/Fermat's Proof | https://proofwiki.org/wiki/Integral_of_Power/Fermat's_Proof | https://proofwiki.org/wiki/Integral_of_Power/Fermat's_Proof | [
"Integral Calculus"
] | [] | [
"Definition:Positive/Integer",
"Definition:Real Number",
"Definition:Subdivision of Interval",
"Definition:Real Interval/Closed",
"Definition:Upper Darboux Sum",
"Sum of Geometric Sequence",
"Definition:Positive/Integer",
"Definition:Strictly Positive",
"Definition:Rational Number",
"Definition:Ra... |
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