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-6400 | Set of Reciprocals of Positive Integers is Nowhere Dense in Reals | Let $N$ be the set defined as:
:$N := \set {\dfrac 1 n: n \in \Z_{>0} }$
where $\Z_{>0}$ is the set of (strictly) positive integers.
Let $\R$ denote the real number line with the usual (Euclidean) metric.
Then $N$ is nowhere dense in $\R$. | From Zero is Limit Point of Integer Reciprocal Space, the only limit point of $N$ is $0$.
Hence:
:$\map \cl N = \set {\dfrac 1 n: n \in \Z_{>0} } \cup \set 0$
where $\map \cl N$ denotes the closure of $N$ in $\R$.
Trivially, $\map \cl N$ contains no open real intervals.
Hence no subset of $\map \cl N$ is open in $\R$.
... | Let $N$ be the [[Definition:Set|set]] defined as:
:$N := \set {\dfrac 1 n: n \in \Z_{>0} }$
where $\Z_{>0}$ is the [[Definition:Set|set]] of [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Let $\R$ denote the [[Definition:Real Number Line with Euclidean Metric|real number line with the usual (Eu... | From [[Zero is Limit Point of Integer Reciprocal Space]], the only [[Definition:Limit Point|limit point]] of $N$ is $0$.
Hence:
:$\map \cl N = \set {\dfrac 1 n: n \in \Z_{>0} } \cup \set 0$
where $\map \cl N$ denotes the [[Definition:Closure (Topology)|closure]] of $N$ in $\R$.
Trivially, $\map \cl N$ contains no [[D... | Set of Reciprocals of Positive Integers is Nowhere Dense in Reals | https://proofwiki.org/wiki/Set_of_Reciprocals_of_Positive_Integers_is_Nowhere_Dense_in_Reals | https://proofwiki.org/wiki/Set_of_Reciprocals_of_Positive_Integers_is_Nowhere_Dense_in_Reals | [
"Real Number Line with Euclidean Metric",
"Integer Reciprocal Space",
"Examples of Nowhere Dense"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Strictly Positive/Integer",
"Definition:Euclidean Metric/Real Number Line",
"Definition:Nowhere Dense"
] | [
"Zero is Limit Point of Integer Reciprocal Space",
"Definition:Limit Point",
"Definition:Closure (Topology)",
"Definition:Real Interval/Open",
"Definition:Subset",
"Definition:Open Set/Topology",
"Definition:Set Union",
"Definition:Subset",
"Definition:Open Set/Topology",
"Definition:Empty Set",
... |
proofwiki-6401 | Nowhere Dense iff Complement of Closure is Everywhere Dense/Corollary | Let $H$ be a closed set of $T$.
Then $H$ is nowhere dense in $T$ {{iff}} $S \setminus H$ is everywhere dense in $T$. | From Closed Set equals its Closure, $H$ is closed in $T$ {{iff}}:
:$H = H^-$
where $H^-$ is the closure of $H$.
The result then follows directly from Nowhere Dense iff Complement of Closure is Everywhere Dense.
{{qed}} | Let $H$ be a [[Definition:Closed Set (Topology)|closed set]] of $T$.
Then $H$ is [[Definition:Nowhere Dense|nowhere dense]] in $T$ {{iff}} $S \setminus H$ is [[Definition:Everywhere Dense|everywhere dense]] in $T$. | From [[Closed Set equals its Closure]], $H$ is [[Definition:Closed Set (Topology)|closed]] in $T$ {{iff}}:
:$H = H^-$
where $H^-$ is the [[Definition:Closure (Topology)|closure]] of $H$.
The result then follows directly from [[Nowhere Dense iff Complement of Closure is Everywhere Dense]].
{{qed}} | Nowhere Dense iff Complement of Closure is Everywhere Dense/Corollary | https://proofwiki.org/wiki/Nowhere_Dense_iff_Complement_of_Closure_is_Everywhere_Dense/Corollary | https://proofwiki.org/wiki/Nowhere_Dense_iff_Complement_of_Closure_is_Everywhere_Dense/Corollary | [
"Nowhere Dense iff Complement of Closure is Everywhere Dense"
] | [
"Definition:Closed Set/Topology",
"Definition:Nowhere Dense",
"Definition:Everywhere Dense"
] | [
"Set is Closed iff Equals Topological Closure",
"Definition:Closed Set/Topology",
"Definition:Closure (Topology)",
"Nowhere Dense iff Complement of Closure is Everywhere Dense"
] |
proofwiki-6402 | Equivalence of Definitions of Topology Generated by Synthetic Basis | Let $S$ be a set.
Let $\BB$ be a synthetic basis on $S$.
{{TFAE|def = Topology Generated by Synthetic Basis}}
=== Definition 1 ===
{{:Definition:Topology Generated by Synthetic Basis/Definition 1}}
=== Definition 2 ===
{{:Definition:Topology Generated by Synthetic Basis/Definition 2}}
=== Definition 3 ===
{{:Definition... | === Definition 1 iff Definition 2 ===
{{:Equivalence of Definitions of Topology Generated by Synthetic Basis/Definition 1 iff Definition 2}}{{qed|lemma}} | Let $S$ be a [[Definition:Set|set]].
Let $\BB$ be a [[Definition:Synthetic Basis|synthetic basis]] on $S$.
{{TFAE|def = Topology Generated by Synthetic Basis}}
=== [[Definition:Topology Generated by Synthetic Basis/Definition 1|Definition 1]] ===
{{:Definition:Topology Generated by Synthetic Basis/Definition 1}}
==... | === [[Equivalence of Definitions of Topology Generated by Synthetic Basis/Definition 1 iff Definition 2|Definition 1 iff Definition 2]] ===
{{:Equivalence of Definitions of Topology Generated by Synthetic Basis/Definition 1 iff Definition 2}}{{qed|lemma}} | Equivalence of Definitions of Topology Generated by Synthetic Basis | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Topology_Generated_by_Synthetic_Basis | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Topology_Generated_by_Synthetic_Basis | [
"Equivalence of Definitions of Topology Generated by Synthetic Basis",
"Topologies Generated by Synthetic Bases"
] | [
"Definition:Set",
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Topology Generated by Synthetic Basis/Definition 1",
"Definition:Topology Generated by Synthetic Basis/Definition 2",
"Definition:Topology Generated by Synthetic Basis/Definition 3"
] | [
"Equivalence of Definitions of Topology Generated by Synthetic Basis/Definition 1 iff Definition 2"
] |
proofwiki-6403 | Union from Synthetic Basis is Topology | Let $\BB$ be a synthetic basis on a set $X$.
Let $\ds \tau = \set {\bigcup \AA: \AA \subseteq \BB}$.
Then $\tau$ is a topology on $X$. | Let $\AA \subseteq \tau$.
It is to be shown that:
:$\ds \bigcup \AA \in \tau$
Define:
:$\ds \AA' = \bigcup_{U \mathop \in \AA} \set {B \in \BB: B \subseteq U}$
By Union is Smallest Superset: Family of Sets, it follows that $\AA' \subseteq \BB$.
Hence, by Equivalence of Definitions of Topology Generated by Synthetic Bas... | Let $\BB$ be a [[Definition:Synthetic Basis|synthetic basis]] on a [[Definition:Set|set]] $X$.
Let $\ds \tau = \set {\bigcup \AA: \AA \subseteq \BB}$.
Then $\tau$ is a [[Definition:Topology|topology]] on $X$. | Let $\AA \subseteq \tau$.
It is to be shown that:
:$\ds \bigcup \AA \in \tau$
Define:
:$\ds \AA' = \bigcup_{U \mathop \in \AA} \set {B \in \BB: B \subseteq U}$
By [[Union is Smallest Superset/Family of Sets|Union is Smallest Superset: Family of Sets]], it follows that $\AA' \subseteq \BB$.
Hence, by [[Equivalence... | Union from Synthetic Basis is Topology/Open Set Axiom 1/Proof 1 | https://proofwiki.org/wiki/Union_from_Synthetic_Basis_is_Topology | https://proofwiki.org/wiki/Union_from_Synthetic_Basis_is_Topology/Open_Set_Axiom_1/Proof_1 | [
"Union from Synthetic Basis is Topology",
"Synthetic Bases"
] | [
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Set",
"Definition:Topology"
] | [
"Union is Smallest Superset/Family of Sets",
"Equivalence of Definitions of Topology Generated by Synthetic Basis",
"Set Union is Self-Distributive/General Result"
] |
proofwiki-6404 | Union from Synthetic Basis is Topology | Let $\BB$ be a synthetic basis on a set $X$.
Let $\ds \tau = \set {\bigcup \AA: \AA \subseteq \BB}$.
Then $\tau$ is a topology on $X$. | Let $U, V \in \tau$.
It is to be shown that:
:$U \cap V \in \tau$
Define:
:$\OO = \set {A \cap B: A, B \in \BB, \, A \subseteq U, \, B \subseteq V}$
By the definition of a synthetic basis:
:$\forall A, B \in \BB: A \cap B \in \tau$
Hence, by the definition of a subset, it follows that $\OO \subseteq \tau$.
By {{Open-se... | Let $\BB$ be a [[Definition:Synthetic Basis|synthetic basis]] on a [[Definition:Set|set]] $X$.
Let $\ds \tau = \set {\bigcup \AA: \AA \subseteq \BB}$.
Then $\tau$ is a [[Definition:Topology|topology]] on $X$. | Let $U, V \in \tau$.
It is to be shown that:
:$U \cap V \in \tau$
Define:
:$\OO = \set {A \cap B: A, B \in \BB, \, A \subseteq U, \, B \subseteq V}$
By the definition of a [[Definition:Synthetic Basis|synthetic basis]]:
:$\forall A, B \in \BB: A \cap B \in \tau$
Hence, by the definition of a [[Definition:Subset|su... | Union from Synthetic Basis is Topology/Open Set Axiom 2/Proof 1 | https://proofwiki.org/wiki/Union_from_Synthetic_Basis_is_Topology | https://proofwiki.org/wiki/Union_from_Synthetic_Basis_is_Topology/Open_Set_Axiom_2/Proof_1 | [
"Union from Synthetic Basis is Topology",
"Synthetic Bases"
] | [
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Set",
"Definition:Topology"
] | [
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Subset",
"Set Intersection Preserves Subsets",
"Union is Smallest Superset/General Result",
"Set is Subset of Union/General Result",
"Definition:Subset",
"Definition:Set Equality/Definition 2"
] |
proofwiki-6405 | Union from Synthetic Basis is Topology | Let $\BB$ be a synthetic basis on a set $X$.
Let $\ds \tau = \set {\bigcup \AA: \AA \subseteq \BB}$.
Then $\tau$ is a topology on $X$. | By the definition of a synthetic basis, $\BB$ is a cover for $S$.
By Equivalent Conditions for Cover by Collection of Subsets, it follows that:
:$\ds S = \bigcup \BB \in \tau$ | Let $\BB$ be a [[Definition:Synthetic Basis|synthetic basis]] on a [[Definition:Set|set]] $X$.
Let $\ds \tau = \set {\bigcup \AA: \AA \subseteq \BB}$.
Then $\tau$ is a [[Definition:Topology|topology]] on $X$. | By the definition of a [[Definition:Synthetic Basis|synthetic basis]], $\BB$ is a [[Definition:Cover of Set|cover]] for $S$.
By [[Equivalent Conditions for Cover by Collection of Subsets]], it follows that:
:$\ds S = \bigcup \BB \in \tau$ | Union from Synthetic Basis is Topology/Open Set Axiom 3/Proof 1 | https://proofwiki.org/wiki/Union_from_Synthetic_Basis_is_Topology | https://proofwiki.org/wiki/Union_from_Synthetic_Basis_is_Topology/Open_Set_Axiom_3/Proof_1 | [
"Union from Synthetic Basis is Topology",
"Synthetic Bases"
] | [
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Set",
"Definition:Topology"
] | [
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Cover of Set",
"Equivalent Conditions for Cover by Collection of Subsets"
] |
proofwiki-6406 | Union from Synthetic Basis is Topology | Let $\BB$ be a synthetic basis on a set $X$.
Let $\ds \tau = \set {\bigcup \AA: \AA \subseteq \BB}$.
Then $\tau$ is a topology on $X$. | By the definition of a synthetic basis, we have that:
:$\forall x \in X: \exists B \in \BB: x \in B \subseteq X$ | Let $\BB$ be a [[Definition:Synthetic Basis|synthetic basis]] on a [[Definition:Set|set]] $X$.
Let $\ds \tau = \set {\bigcup \AA: \AA \subseteq \BB}$.
Then $\tau$ is a [[Definition:Topology|topology]] on $X$. | By the definition of a [[Definition:Synthetic Basis|synthetic basis]], we have that:
:$\forall x \in X: \exists B \in \BB: x \in B \subseteq X$ | Union from Synthetic Basis is Topology/Open Set Axiom 3/Proof 2 | https://proofwiki.org/wiki/Union_from_Synthetic_Basis_is_Topology | https://proofwiki.org/wiki/Union_from_Synthetic_Basis_is_Topology/Open_Set_Axiom_3/Proof_2 | [
"Union from Synthetic Basis is Topology",
"Synthetic Bases"
] | [
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Set",
"Definition:Topology"
] | [
"Definition:Basis (Topology)/Synthetic Basis"
] |
proofwiki-6407 | Union from Synthetic Basis is Topology | Let $\BB$ be a synthetic basis on a set $X$.
Let $\ds \tau = \set {\bigcup \AA: \AA \subseteq \BB}$.
Then $\tau$ is a topology on $X$. | This theorem is divided into the following sections, each one of which contains its subproof under the associated links.
=== Open Set Axiom $(\text O 1)$: Union of Open Sets ===
{{:Union from Synthetic Basis is Topology/Open Set Axiom 1}}
=== Open Set Axiom $(\text O 2)$: Pairwise Intersection of Open Sets ===
{{:Union... | Let $\BB$ be a [[Definition:Synthetic Basis|synthetic basis]] on a [[Definition:Set|set]] $X$.
Let $\ds \tau = \set {\bigcup \AA: \AA \subseteq \BB}$.
Then $\tau$ is a [[Definition:Topology|topology]] on $X$. | This theorem is divided into the following sections, each one of which contains its subproof under the associated links.
=== [[Union from Synthetic Basis is Topology/Open Set Axiom 1|Open Set Axiom $(\text O 1)$: Union of Open Sets]] ===
{{:Union from Synthetic Basis is Topology/Open Set Axiom 1}}
=== [[Union from S... | Union from Synthetic Basis is Topology/Proof | https://proofwiki.org/wiki/Union_from_Synthetic_Basis_is_Topology | https://proofwiki.org/wiki/Union_from_Synthetic_Basis_is_Topology/Proof | [
"Union from Synthetic Basis is Topology",
"Synthetic Bases"
] | [
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Set",
"Definition:Topology"
] | [
"Union from Synthetic Basis is Topology/Open Set Axiom 1",
"Union from Synthetic Basis is Topology/Open Set Axiom 2",
"Union from Synthetic Basis is Topology/Open Set Axiom 3"
] |
proofwiki-6408 | Binary Product as Limit | Let $\mathbf C$ be a metacategory.
Let $C_1, C_2$ be objects of $\mathbf C$.
Let their binary product $C_1 \times C_2$ exist in $\mathbf C$.
Then $C_1 \times C_2$ is the limit of the diagram $D: 2 \to \mathbf C$ defined by:
:$D_0 := C_1, D_1 := C_2$
where $2$ is the discrete category with two objects $0, 1$. | Since there are no non-identity morphisms, a cone to $D$ is simply a pair:
::<nowiki>$\begin{xy}\xymatrix@+1em@L+3px{
C_1
&
C
\ar[l]_*+{f_1}
\ar[r]^*+{f_2}
&
C_2
}\end{xy}$</nowiki>
of morphisms with common domain $C$.
By the universal mapping property of the binary product $C_1 \times C_2$, for such a cone to $... | Let $\mathbf C$ be a [[Definition:Metacategory|metacategory]].
Let $C_1, C_2$ be [[Definition:Object (Category Theory)|objects]] of $\mathbf C$.
Let their [[Definition:Binary Product (Category Theory)|binary product]] $C_1 \times C_2$ exist in $\mathbf C$.
Then $C_1 \times C_2$ is the [[Definition:Limit (Category T... | Since there are no non-[[Definition:Identity Morphism|identity morphisms]], a [[Definition:Cone (Category Theory)|cone]] to $D$ is simply a pair:
::<nowiki>$\begin{xy}\xymatrix@+1em@L+3px{
C_1
&
C
\ar[l]_*+{f_1}
\ar[r]^*+{f_2}
&
C_2
}\end{xy}$</nowiki>
of [[Definition:Morphism (Category Theory)|morphisms]] wit... | Binary Product as Limit | https://proofwiki.org/wiki/Binary_Product_as_Limit | https://proofwiki.org/wiki/Binary_Product_as_Limit | [
"Category Theory",
"Limits and Colimits"
] | [
"Definition:Metacategory",
"Definition:Object (Category Theory)",
"Definition:Product (Category Theory)/Binary Product",
"Definition:Limit (Category Theory)",
"Definition:Diagram (Category Theory)",
"Definition:Discrete Category",
"Definition:Object (Category Theory)"
] | [
"Definition:Identity Morphism",
"Definition:Cone (Category Theory)",
"Definition:Morphism",
"Definition:Domain (Category Theory)",
"Definition:Product UMP (Category Theory)",
"Definition:Product (Category Theory)/Binary Product",
"Definition:Cone (Category Theory)",
"Definition:Unique",
"Definition:... |
proofwiki-6409 | Equalizer as Limit | Let $\mathbf C$ be a metacategory.
Let $f_1, f_2: C_1 \to C_2$ be morphisms of $\mathbf C$.
Let their equalizer $e: E \to C_1$ exist in $\mathbf C$.
Define $\tilde e : E \to C_2$ by:
:$\tilde e = f_1 e = f_2 e$
Then $\struct {E, \sequence {e, \tilde e} }$ is the limit of the diagram $D: \mathbf J \to \mathbf C$ defined... | Let $\struct {A, \sequence {a, \tilde a} }$ be a cone to $D$.
That is:
::<nowiki>$\begin{xy}\xymatrix{
A
\ar[d]_*+{a}
\ar[dr]^*+{\tilde a}
\\
C_1
\ar[r]<2pt>^*+{f_1}
\ar[r]<-2pt>_*+{f_2}
&
C_2
}\end{xy}$</nowiki>
is a commutative diagram.
That is, $f_1 a = f_2 a$.
By {{Defof|Equalizer}}, there is a unique $u... | Let $\mathbf C$ be a [[Definition:Metacategory|metacategory]].
Let $f_1, f_2: C_1 \to C_2$ be [[Definition:Morphism (Category Theory)|morphisms]] of $\mathbf C$.
Let their [[Definition:Equalizer|equalizer]] $e: E \to C_1$ exist in $\mathbf C$.
Define $\tilde e : E \to C_2$ by:
:$\tilde e = f_1 e = f_2 e$
Then $\st... | Let $\struct {A, \sequence {a, \tilde a} }$ be a [[Definition:Cone (Category Theory)|cone]] to $D$.
That is:
::<nowiki>$\begin{xy}\xymatrix{
A
\ar[d]_*+{a}
\ar[dr]^*+{\tilde a}
\\
C_1
\ar[r]<2pt>^*+{f_1}
\ar[r]<-2pt>_*+{f_2}
&
C_2
}\end{xy}$</nowiki>
is a [[Definition:Commutative Diagram|commutative diagra... | Equalizer as Limit | https://proofwiki.org/wiki/Equalizer_as_Limit | https://proofwiki.org/wiki/Equalizer_as_Limit | [
"Category Theory",
"Limits and Colimits"
] | [
"Definition:Metacategory",
"Definition:Morphism",
"Definition:Equalizer",
"Definition:Limit (Category Theory)",
"Definition:Diagram (Category Theory)"
] | [
"Definition:Cone (Category Theory)",
"Definition:Commutative Diagram",
"Definition:Unique",
"Definition:Commutative Diagram",
"Definition:Unique",
"Definition:Commutative Diagram"
] |
proofwiki-6410 | Terminal Object as Limit | Let $\mathbf C$ be a metacategory.
Let $\mathbf C$ have a terminal object $1$.
Then $1$ is the limit of the unique diagram $D: \mathbf 0 \to \mathbf C$, where $\mathbf 0$ is the zero category. | By definition of cone to $D$, the objects of $\map {\mathbf{Cone} } D$ are just the objects of $\mathbf C$ with no associated morphisms.
In particular, the morphisms of $\map {\mathbf{Cone} } D$ are just the morphisms of $\mathbf C$.
Thus $1$ is a terminal object in $\map {\mathbf{Cone} } D$.
That is, $1$ is the limit ... | Let $\mathbf C$ be a [[Definition:Metacategory|metacategory]].
Let $\mathbf C$ have a [[Definition:Terminal Object|terminal object]] $1$.
Then $1$ is the [[Definition:Limit (Category Theory)|limit]] of the [[Definition:Unique|unique]] [[Definition:Diagram (Category Theory)|diagram]] $D: \mathbf 0 \to \mathbf C$, whe... | By [[Definition:Cone (Category Theory)|definition of cone to $D$]], the [[Definition:Object (Category Theory)|objects]] of $\map {\mathbf{Cone} } D$ are just the [[Definition:Object (Category Theory)|objects]] of $\mathbf C$ with no associated [[Definition:Morphism (Category Theory)|morphisms]].
In particular, the [[D... | Terminal Object as Limit | https://proofwiki.org/wiki/Terminal_Object_as_Limit | https://proofwiki.org/wiki/Terminal_Object_as_Limit | [
"Objects (Category Theory)",
"Limits and Colimits"
] | [
"Definition:Metacategory",
"Definition:Terminal Object",
"Definition:Limit (Category Theory)",
"Definition:Unique",
"Definition:Diagram (Category Theory)",
"Definition:Zero (Category)"
] | [
"Definition:Cone (Category Theory)",
"Definition:Object (Category Theory)",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Morphism of Cones",
"Definition:Morphism",
"Definition:Terminal Object",
"Definition:Limit (Category Theory)"
] |
proofwiki-6411 | Pullback as Limit | Let $\mathbf C$ be a metacategory.
Let $f_1: A \to C$ and $f_2: B \to C$ be morphisms of $\mathbf C$.
Let their pullback:
::<nowiki>$\begin{xy}\xymatrix@+1em@L+3px{
P
\ar[r]^*+{p_1}
\ar[d]_*+{p_2}
&
A
\ar[d]^*+{f_1}
\\
B
\ar[r]_*+{f_2}
&
C
}\end{xy}$</nowiki>
exist in $\mathbf C$.
Define the morphisms $p_3 ... | First, $\struct {P, \sequence {p_1, p_2, p_3} }$ is indeed a cone to $D$, as:
::<nowiki>$\begin{xy}\xymatrix@+1em@L+3px{
P
\ar[r]^*+{p_1}
\ar[d]_*+{p_2}
\ar[rd]^*+{p_3}
&
A
\ar[d]^*+{f_1}
\\
B
\ar[r]_*+{f_2}
&
C
}\end{xy}$</nowiki>
is a commutative diagram.
We need to show that this is a terminal object i... | Let $\mathbf C$ be a [[Definition:Metacategory|metacategory]].
Let $f_1: A \to C$ and $f_2: B \to C$ be [[Definition:Morphism (Category Theory)|morphisms]] of $\mathbf C$.
Let their [[Definition:Pullback (Category Theory)|pullback]]:
::<nowiki>$\begin{xy}\xymatrix@+1em@L+3px{
P
\ar[r]^*+{p_1}
\ar[d]_*+{p_2}
&
... | First, $\struct {P, \sequence {p_1, p_2, p_3} }$ is indeed a [[Definition:Cone (Category Theory)|cone]] to $D$, as:
::<nowiki>$\begin{xy}\xymatrix@+1em@L+3px{
P
\ar[r]^*+{p_1}
\ar[d]_*+{p_2}
\ar[rd]^*+{p_3}
&
A
\ar[d]^*+{f_1}
\\
B
\ar[r]_*+{f_2}
&
C
}\end{xy}$</nowiki>
is a [[Definition:Commutative Dia... | Pullback as Limit | https://proofwiki.org/wiki/Pullback_as_Limit | https://proofwiki.org/wiki/Pullback_as_Limit | [
"Limits and Colimits",
"Pullbacks"
] | [
"Definition:Metacategory",
"Definition:Morphism",
"Definition:Pullback (Category Theory)",
"Definition:Morphism",
"Definition:Limit (Category Theory)",
"Definition:Diagram (Category Theory)"
] | [
"Definition:Cone (Category Theory)",
"Definition:Commutative Diagram",
"Definition:Terminal Object",
"Definition:Cone (Category Theory)",
"Definition:Commutative Diagram",
"Definition:Commutative Diagram",
"Definition:Morphism of Cones",
"Definition:Morphism of Cones",
"Definition:Terminal Object"
] |
proofwiki-6412 | Identification Topology is Topology | Let $T_1 = \struct {S_1, \tau_1}$ be a topological space.
Let $S_2$ be a set.
Let $f: S_1 \to S_2$ be a mapping.
Let $\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $\struct {S_1, \tau_1}$.
Then $\tau_2$ is a topology on $S_2$. | By definition:
:$\tau_2 = \set {V \in \powerset {S_2}: f^{-1} \sqbrk V \in \tau_1}$
We examine each of the open set axioms in turn: | Let $T_1 = \struct {S_1, \tau_1}$ be a [[Definition:Topological Space|topological space]].
Let $S_2$ be a [[Definition:Set|set]].
Let $f: S_1 \to S_2$ be a [[Definition:Mapping|mapping]].
Let $\tau_2$ be the [[Definition:Identification Topology|identification topology on $S_2$ with respect to $f$ and $\struct {S_1, ... | By definition:
:$\tau_2 = \set {V \in \powerset {S_2}: f^{-1} \sqbrk V \in \tau_1}$
We examine each of the [[Axiom:Open Set Axioms|open set axioms]] in turn: | Identification Topology is Topology | https://proofwiki.org/wiki/Identification_Topology_is_Topology | https://proofwiki.org/wiki/Identification_Topology_is_Topology | [
"Identification Topology"
] | [
"Definition:Topological Space",
"Definition:Set",
"Definition:Mapping",
"Definition:Identification Topology",
"Definition:Topology"
] | [
"Axiom:Open Set Axioms",
"Axiom:Open Set Axioms"
] |
proofwiki-6413 | Identification Mapping is Continuous | Let $T_1 = \struct {S_1, \tau_1}$ be a topological space.
Let $S_2$ be a set.
Let $f: S_1 \to S_2$ be a mapping.
Let $\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $\struct {S_1, \tau_1}$.
Then the identification mapping $f$ is continuous. | By definition of identification topology:
:$\tau_2 = \set {V \in \powerset {S_2}: f^{-1} \sqbrk V \in \tau_1}$
That is:
:$V \in \tau_2 \implies f^{-1} \sqbrk V \in \tau_1$
This is precisely the definition of a continuous mapping.
{{qed}}
Category:Identification Topology
p1p49egjbhzpe3wej3gre4srnw99xg3 | Let $T_1 = \struct {S_1, \tau_1}$ be a [[Definition:Topological Space|topological space]].
Let $S_2$ be a [[Definition:Set|set]].
Let $f: S_1 \to S_2$ be a [[Definition:Mapping|mapping]].
Let $\tau_2$ be the [[Definition:Identification Topology|identification topology on $S_2$ with respect to $f$ and $\struct {S_1, ... | By definition of [[Definition:Identification Topology|identification topology]]:
:$\tau_2 = \set {V \in \powerset {S_2}: f^{-1} \sqbrk V \in \tau_1}$
That is:
:$V \in \tau_2 \implies f^{-1} \sqbrk V \in \tau_1$
This is precisely the definition of a [[Definition:Continuous Mapping (Topology)|continuous mapping]].
{{qe... | Identification Mapping is Continuous | https://proofwiki.org/wiki/Identification_Mapping_is_Continuous | https://proofwiki.org/wiki/Identification_Mapping_is_Continuous | [
"Identification Topology"
] | [
"Definition:Topological Space",
"Definition:Set",
"Definition:Mapping",
"Definition:Identification Topology",
"Definition:Identification Topology/Identification Mapping",
"Definition:Continuous Mapping (Topology)"
] | [
"Definition:Identification Topology",
"Definition:Continuous Mapping (Topology)",
"Category:Identification Topology"
] |
proofwiki-6414 | Existence and Uniqueness of Identification Topology | Let $T_1 = \struct {S_1, \tau_1}$ be a topological space.
Let $S_2$ be a set.
Let $f: S_1 \to S_2$ be a mapping.
Let $\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $\struct {S_1, \tau_1}$.
Then the identification topology on $S_2$ with respect to $f$ and $\struct {S_1, \tau_1}$ always exists a... | Let $\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $\struct {S_1, \tau_1}$.
By definition:
:$\tau_2 := \set {V \in \powerset {S_2}: f^{-1} \sqbrk V \in \tau_1} \subseteq \powerset {S_2}$
where $\powerset {S_2}$ is the power set of $S_2$.
Let $V \subseteq S_2$.
Then either:
:$f^{-1} \sqbrk V \i... | Let $T_1 = \struct {S_1, \tau_1}$ be a [[Definition:Topological Space|topological space]].
Let $S_2$ be a [[Definition:Set|set]].
Let $f: S_1 \to S_2$ be a [[Definition:Mapping|mapping]].
Let $\tau_2$ be the [[Definition:Identification Topology|identification topology on $S_2$ with respect to $f$ and $\struct {S_1, ... | Let $\tau_2$ be the [[Definition:Identification Topology|identification topology]] on $S_2$ with respect to $f$ and $\struct {S_1, \tau_1}$.
By definition:
:$\tau_2 := \set {V \in \powerset {S_2}: f^{-1} \sqbrk V \in \tau_1} \subseteq \powerset {S_2}$
where $\powerset {S_2}$ is the [[Definition:Power Set|power set]] o... | Existence and Uniqueness of Identification Topology | https://proofwiki.org/wiki/Existence_and_Uniqueness_of_Identification_Topology | https://proofwiki.org/wiki/Existence_and_Uniqueness_of_Identification_Topology | [
"Identification Topology"
] | [
"Definition:Topological Space",
"Definition:Set",
"Definition:Mapping",
"Definition:Identification Topology",
"Definition:Identification Topology",
"Definition:Unique"
] | [
"Definition:Identification Topology",
"Definition:Power Set",
"Definition:Set",
"Definition:Set",
"Definition:Element"
] |
proofwiki-6415 | Category has Finite Limits iff Finite Products and Equalizers | Let $\mathbf C$ be a metacategory.
Then:
:$\mathbf C$ has all finite limits
{{iff}}:
:$\mathbf C$ has all finite products and equalizers. | === Necessary Condition ===
By definition, finite products are instances of finite limits.
So are equalizers, by Equalizer as Limit.
{{qed|lemma}} | Let $\mathbf C$ be a [[Definition:Metacategory|metacategory]].
Then:
:$\mathbf C$ has all [[Definition:Finite Limit (Category Theory)|finite limits]]
{{iff}}:
:$\mathbf C$ has all [[Definition:Product (Category Theory)|finite products]] and [[Definition:Equalizer|equalizers]]. | === Necessary Condition ===
By definition, [[Definition:Finite Product (Category Theory)|finite products]] are instances of [[Definition:Finite Limit (Category Theory)|finite limits]].
So are [[Definition:Equalizer|equalizers]], by [[Equalizer as Limit]].
{{qed|lemma}} | Category has Finite Limits iff Finite Products and Equalizers | https://proofwiki.org/wiki/Category_has_Finite_Limits_iff_Finite_Products_and_Equalizers | https://proofwiki.org/wiki/Category_has_Finite_Limits_iff_Finite_Products_and_Equalizers | [
"Products (Category Theory)"
] | [
"Definition:Metacategory",
"Definition:Limit (Category Theory)/Finite Limit",
"Definition:Product (Category Theory)",
"Definition:Equalizer"
] | [
"Definition:Product (Category Theory)/General Definition/Finite Product",
"Definition:Limit (Category Theory)/Finite Limit",
"Definition:Equalizer",
"Equalizer as Limit",
"Definition:Product (Category Theory)/General Definition/Finite Product",
"Definition:Equalizer",
"Definition:Equalizer",
"Definiti... |
proofwiki-6416 | Identification Topology is Finest Topology for Mapping to be Continuous | Let $T_1 := \struct {S_1, \tau_1}$ be a topological space.
Let $S_2$ be a set.
Let $f: S_1 \to S_2$ be a mapping.
Let $\tau_2$ be the identification topology on $S_2$ with respect to $f$ and $\struct {S_1, \tau_1}$.
Let $T_2 := \struct {S_2, \tau_2}$ be the resulting topological space.
Then $\tau_2$ is the finest topol... | It is established in Identification Mapping is Continuous that $f$ is continuous.
Let $\tau \subseteq \powerset {S_2}$ be a topology on $S_2$ for which $f$ is $\struct {\tau_1, \tau}$-continuous.
Let $U \in \tau$.
Then by definition of $\struct {\tau_1, \tau}$-continuous:
:$f^{-1} \sqbrk U \in \tau_1$
By definition of ... | Let $T_1 := \struct {S_1, \tau_1}$ be a [[Definition:Topological Space|topological space]].
Let $S_2$ be a [[Definition:Set|set]].
Let $f: S_1 \to S_2$ be a [[Definition:Mapping|mapping]].
Let $\tau_2$ be the [[Definition:Identification Topology|identification topology on $S_2$ with respect to $f$ and $\struct {S_1,... | It is established in [[Identification Mapping is Continuous]] that $f$ is [[Definition:Everywhere Continuous Mapping (Topology)|continuous]].
Let $\tau \subseteq \powerset {S_2}$ be a [[Definition:Topology|topology]] on $S_2$ for which $f$ is [[Definition:Everywhere Continuous Mapping (Topology)|$\struct {\tau_1, \ta... | Identification Topology is Finest Topology for Mapping to be Continuous | https://proofwiki.org/wiki/Identification_Topology_is_Finest_Topology_for_Mapping_to_be_Continuous | https://proofwiki.org/wiki/Identification_Topology_is_Finest_Topology_for_Mapping_to_be_Continuous | [
"Identification Topology",
"Finer Topology"
] | [
"Definition:Topological Space",
"Definition:Set",
"Definition:Mapping",
"Definition:Identification Topology",
"Definition:Topological Space",
"Definition:Finer Topology",
"Definition:Continuous Mapping (Topology)/Everywhere"
] | [
"Identification Mapping is Continuous",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Topology",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Subset",
"Definition:Finer Topology",
"Definition:Topology",
... |
proofwiki-6417 | Identification Topology equals Quotient Topology on Induced Equivalence | Let $\struct {S_1, \tau_1}$ be a topological space.
Let $S_2$ be a set.
Let $f: S_1 \to S_2$ be a surjective mapping.
Let $\tau_2$ be the identification topology on $S_2$ {{WRT}} $f$ and $\struct {S_1, \tau_1}$:
:$\tau_2 = \set {V \subseteq S_2 : f^{-1} \sqbrk V \in \tau_1}$
Let $\RR_f \subseteq S_1 \times S_1$ be the ... | Define a mapping $\tilde f: S_1 / \RR_f \to S_2$ by:
:$\map {\tilde f} {\eqclass s { \RR_f} } = \map f s$
:<nowiki>$\begin{xy} \xymatrix@L+2mu@+1em{
S_1 \ar[r]^*{f}
\ar[rd]_*{ q_{\RR_f} }
&
S_2
\\ &
S_1 / \RR_f \ar@{-->}[u]^*{\tilde f}
}\end{xy}$</nowiki>
Then $\tilde f$ is well-defined, as for all $s' \in \eq... | Let $\struct {S_1, \tau_1}$ be a [[Definition:Topological Space|topological space]].
Let $S_2$ be a [[Definition:Set|set]].
Let $f: S_1 \to S_2$ be a [[Definition:Surjection|surjective mapping]].
Let $\tau_2$ be the [[Definition:Identification Topology|identification topology on $S_2$ {{WRT}} $f$ and $\struct {S_1, ... | Define a [[Definition:Mapping|mapping]] $\tilde f: S_1 / \RR_f \to S_2$ by:
:$\map {\tilde f} {\eqclass s { \RR_f} } = \map f s$
:<nowiki>$\begin{xy} \xymatrix@L+2mu@+1em{
S_1 \ar[r]^*{f}
\ar[rd]_*{ q_{\RR_f} }
&
S_2
\\ &
S_1 / \RR_f \ar@{-->}[u]^*{\tilde f}
}\end{xy}$</nowiki>
Then $\tilde f$ is well-defin... | Identification Topology equals Quotient Topology on Induced Equivalence | https://proofwiki.org/wiki/Identification_Topology_equals_Quotient_Topology_on_Induced_Equivalence | https://proofwiki.org/wiki/Identification_Topology_equals_Quotient_Topology_on_Induced_Equivalence | [
"Identification Topology",
"Quotient Topologies"
] | [
"Definition:Topological Space",
"Definition:Set",
"Definition:Surjection",
"Definition:Identification Topology",
"Definition:Equivalence Relation Induced by Mapping",
"Definition:Quotient Mapping",
"Definition:Quotient Topology",
"Definition:Homeomorphism",
"Definition:Quotient Topology/Quotient Spa... | [
"Definition:Mapping",
"Definition:Injection",
"Definition:Surjection",
"Definition:Surjection",
"Identification Mapping is Continuous",
"Definition:Quotient Topology",
"Definition:Continuous Mapping (Topology)",
"Definition:Continuous Mapping (Topology)",
"Definition:Quotient Topology",
"Definitio... |
proofwiki-6418 | Young's Inequality for Increasing Functions | Let $a_0$ and $b_0$ be strictly positive real numbers.
Let $f: \closedint 0 {a_0} \to \closedint 0 {b_0}$ be a strictly increasing bijection.
Let $a$ and $b$ be real numbers such that $0 \le a \le a_0$ and $0 \le b \le b_0$.
Then:
:$\ds ab \le \int_0^a \map f u \rd u + \int_0^b \map {f^{-1} } v \rd v$
where $\ds \int$ ... | 200pxthumbrightThe blue colored region corresponds to $\ds \int_0^a \map f u \rd u$ and the red colored region to $\ds \int_0^b \map {f^{-1} } v \rd v$.
{{ProofWanted}}
{{Namedfor|William Henry Young|cat = Young, W.H.}} | Let $a_0$ and $b_0$ be [[Definition:Strictly Positive Real Number|strictly positive real numbers]].
Let $f: \closedint 0 {a_0} \to \closedint 0 {b_0}$ be a [[Definition:Strictly Increasing Real Function|strictly increasing]] [[Definition:Bijection|bijection]].
Let $a$ and $b$ be [[Definition:Real Number|real numbers]... | [[File:Young's Ineq for Increasing Functions.png|200px|thumb|right|The blue colored region corresponds to $\ds \int_0^a \map f u \rd u$ and the red colored region to $\ds \int_0^b \map {f^{-1} } v \rd v$.]]
{{ProofWanted}}
{{Namedfor|William Henry Young|cat = Young, W.H.}} | Young's Inequality for Increasing Functions | https://proofwiki.org/wiki/Young's_Inequality_for_Increasing_Functions | https://proofwiki.org/wiki/Young's_Inequality_for_Increasing_Functions | [
"Young's Inequality for Increasing Functions",
"Integral Calculus",
"Strictly Increasing Real Functions"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Strictly Increasing/Real Function",
"Definition:Bijection",
"Definition:Real Number",
"Definition:Definite Integral"
] | [
"File:Young's Ineq for Increasing Functions.png"
] |
proofwiki-6419 | Set is Closed iff Equals Topological Closure | Let $T$ be a topological space.
Let $H \subseteq T$.
Then $H$ is closed in $T$ {{iff}}:
:$H = \map \cl H$ | Let $H'$ denote the derived set of $H$.
By Closed Set iff Contains all its Limit Points, $H$ is closed in $T$ {{iff}} $H' \subseteq H$.
By Union with Superset is Superset, $H' \subseteq H$ {{iff}} $H = H \cup H'$.
The result follows from the definition of closure.
{{qed}} | Let $T$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq T$.
Then $H$ is [[Definition:Closed Set (Topology)|closed]] in $T$ {{iff}}:
:$H = \map \cl H$ | Let $H'$ denote the [[Definition:Derived Set|derived set]] of $H$.
By [[Closed Set iff Contains all its Limit Points]], $H$ is [[Definition:Closed Set (Topology)|closed]] in $T$ {{iff}} $H' \subseteq H$.
By [[Union with Superset is Superset]], $H' \subseteq H$ {{iff}} $H = H \cup H'$.
The result follows from the de... | Set is Closed iff Equals Topological Closure/Proof 1 | https://proofwiki.org/wiki/Set_is_Closed_iff_Equals_Topological_Closure | https://proofwiki.org/wiki/Set_is_Closed_iff_Equals_Topological_Closure/Proof_1 | [
"Set is Closed iff Equals Topological Closure",
"Closed Sets",
"Set Closures"
] | [
"Definition:Topological Space",
"Definition:Closed Set/Topology"
] | [
"Definition:Derived Set",
"Equivalence of Definitions of Closed Set",
"Definition:Closed Set/Topology",
"Union with Superset is Superset",
"Definition:Closure (Topology)/Definition 1"
] |
proofwiki-6420 | Set is Closed iff Equals Topological Closure | Let $T$ be a topological space.
Let $H \subseteq T$.
Then $H$ is closed in $T$ {{iff}}:
:$H = \map \cl H$ | Let $H^\complement$ denote the relative complement of $H$ in $T$.
By definition, we have that $H$ is closed in $T$ {{iff}} $H^\complement$ is open in $T$.
By Set is Open iff Neighborhood of all its Points, this is equivalent to:
:$\forall x \in H^\complement: \exists U \in \tau: x \in U \subseteq H^\complement$
By Empt... | Let $T$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq T$.
Then $H$ is [[Definition:Closed Set (Topology)|closed]] in $T$ {{iff}}:
:$H = \map \cl H$ | Let $H^\complement$ denote the [[Definition:Relative Complement|relative complement]] of $H$ in $T$.
By definition, we have that $H$ is [[Definition:Closed Set (Topology)|closed]] in $T$ {{iff}} $H^\complement$ is [[Definition:Open Set (Topology)|open]] in $T$.
By [[Set is Open iff Neighborhood of all its Points]], ... | Set is Closed iff Equals Topological Closure/Proof 2 | https://proofwiki.org/wiki/Set_is_Closed_iff_Equals_Topological_Closure | https://proofwiki.org/wiki/Set_is_Closed_iff_Equals_Topological_Closure/Proof_2 | [
"Set is Closed iff Equals Topological Closure",
"Closed Sets",
"Set Closures"
] | [
"Definition:Topological Space",
"Definition:Closed Set/Topology"
] | [
"Definition:Relative Complement",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Set is Open iff Neighborhood of all its Points",
"Definition:Logical Equivalence",
"Empty Intersection iff Subset of Complement",
"Condition for Point being in Closure",
"Definition:Open Set/Topology",... |
proofwiki-6421 | Equivalence of Definitions of Compact Topological Subspace | {{TFAE|def = Compact Topological Subspace}}
Let $T = \struct {S, \tau}$ be a topological space.
Let $T_H = \struct {H, \tau_H}$ be a topological subspace of $T$, where $H \subseteq S$. | === $1$ implies $2$ ===
Suppose $T_H$ is compact in the sense of Definition 1.
Let $\CC$ be a cover of $H$ by open sets of $T$.
Then for each $U \in \CC$, $U \cap H$ is open in $T_H$ by definition of the subspace topology.
Since $\CC$ is a cover of $H$ it follows that $\CC' = \set{U \cap H: U \in \CC}$ is also a cover ... | {{TFAE|def = Compact Topological Subspace}}
Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T_H = \struct {H, \tau_H}$ be a [[Definition:Topological Subspace|topological subspace]] of $T$, where $H \subseteq S$. | === $1$ implies $2$ ===
Suppose $T_H$ is compact in the sense of [[Definition:Compact Topological Subspace/Definition 1|Definition 1]].
Let $\CC$ be a [[Definition:Cover of Set|cover]] of $H$ by [[Definition:Open Set (Topology)|open sets]] of $T$.
Then for each $U \in \CC$, $U \cap H$ is [[Definition:Open Set (Topol... | Equivalence of Definitions of Compact Topological Subspace | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Compact_Topological_Subspace | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Compact_Topological_Subspace | [
"Compact Topological Spaces"
] | [
"Definition:Topological Space",
"Definition:Topological Subspace"
] | [
"Definition:Compact Topological Subspace/Definition 1",
"Definition:Cover of Set",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Topological Subspace",
"Definition:Cover of Set",
"Definition:Cover of Set",
"Definition:By Hypothesis",
"Definition:Subcover/Finite",
"Def... |
proofwiki-6422 | Covariant Hom Functor is Continuous | Let $\mathbf{Set}$ be the category of sets.
Let $\mathbf C$ be a locally small category.
Let $C$ be an object of $\mathbf C$, and let $\map {\operatorname{Hom}_{\mathbf C} } {C, \cdot}: \mathbf C \to \mathbf{Set}$ be the covariant hom functor based at $C$.
Then $\map {\operatorname{Hom}_{\mathbf C} } {C, \cdot}$ is a c... | Let $F := \map {\operatorname{Hom}_{\mathbf C} } {C, \cdot}$.
Let $D: \mathbf J \to \mathbf C$ be a diagram.
Suppose the cone $\paren { L, \sequence {p_j} }$ is a limit for $D$.
We need to show that $\paren { F L, \sequence {F p_j } }$ is a limit for the diagram $F D: \mathbf J \to \mathbf{Set}$.
As $F$ is a covariant... | Let $\mathbf{Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $\mathbf C$ be a [[Definition:Locally Small Category|locally small category]].
Let $C$ be an [[Definition:Object|object]] of $\mathbf C$, and let $\map {\operatorname{Hom}_{\mathbf C} } {C, \cdot}: \mathbf C \to \mathbf{Set}$ be the [[Def... | Let $F := \map {\operatorname{Hom}_{\mathbf C} } {C, \cdot}$.
Let $D: \mathbf J \to \mathbf C$ be a [[Definition:Diagram (Category Theory)|diagram]].
Suppose the [[Definition:Cone (Category Theory)|cone]] $\paren { L, \sequence {p_j} }$ is a [[Definition:Limit (Category Theory)|limit]] for $D$.
We need to show that... | Covariant Hom Functor is Continuous | https://proofwiki.org/wiki/Covariant_Hom_Functor_is_Continuous | https://proofwiki.org/wiki/Covariant_Hom_Functor_is_Continuous | [
"Functors"
] | [
"Definition:Category of Sets",
"Definition:Locally Small Category",
"Definition:Object",
"Definition:Covariant Hom Functor",
"Definition:Continuous Functor"
] | [
"Definition:Diagram (Category Theory)",
"Definition:Cone (Category Theory)",
"Definition:Limit (Category Theory)",
"Definition:Limit (Category Theory)",
"Definition:Diagram (Category Theory)",
"Definition:Covariant Hom Functor",
"Definition:Cone (Category Theory)",
"Definition:Terminal Object",
"Def... |
proofwiki-6423 | Composite Mapping is Mapping | Let $S_1, S_2, S_3$ be sets.
Let $f: S_1 \to S_2$ and $g: S_2 \to S_3$ be mappings.
Then the composite mapping $g \circ f$ is also a mapping. | The composite of $f$ and $g$ is defined as:
:$g \circ f := \set {\tuple {x, z} \in S_1 \times S_3: \exists y \in S_2: \tuple {x, y} \in f \land \tuple {y, z} \in g}$
It is necessary to show that $g \circ f$ is both left-total and many-to-one. | Let $S_1, S_2, S_3$ be [[Definition:Set|sets]].
Let $f: S_1 \to S_2$ and $g: S_2 \to S_3$ be [[Definition:Mapping|mappings]].
Then the [[Definition:Composite Mapping|composite mapping]] $g \circ f$ is also a [[Definition:Mapping|mapping]]. | The [[Definition:Composite Mapping|composite of $f$ and $g$]] is defined as:
:$g \circ f := \set {\tuple {x, z} \in S_1 \times S_3: \exists y \in S_2: \tuple {x, y} \in f \land \tuple {y, z} \in g}$
It is necessary to show that $g \circ f$ is both [[Definition:Left-Total Relation|left-total]] and [[Definition:Many-t... | Composite Mapping is Mapping | https://proofwiki.org/wiki/Composite_Mapping_is_Mapping | https://proofwiki.org/wiki/Composite_Mapping_is_Mapping | [
"Composite Mappings"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Composition of Mappings",
"Definition:Mapping"
] | [
"Definition:Composition of Mappings",
"Definition:Left-Total Relation",
"Definition:Many-to-One Relation",
"Definition:Left-Total Relation",
"Definition:Many-to-One Relation",
"Definition:Many-to-One Relation"
] |
proofwiki-6424 | Normal Space is Regular | Let $\struct {S, \tau}$ be a normal space.
Then $\struct {S, \tau}$ is also a regular space. | Let $T = \struct {S, \tau}$ be a normal space.
From Normal Space is $T_3$, we have that $T$ is a $T_3$ space.
We also have by definition of normal space that $T$ is a $T_1$ space.
From $T_1$ Space is $T_0$ we have that $T$ is a $T_0$ space
So $T$ is both a $T_3$ space and a $T_0$ space.
Hence $T$ is a regular space by ... | Let $\struct {S, \tau}$ be a [[Definition:Normal Space|normal space]].
Then $\struct {S, \tau}$ is also a [[Definition:Regular Space|regular space]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Normal Space|normal space]].
From [[Normal Space is T3|Normal Space is $T_3$]], we have that $T$ is a [[Definition:T3 Space|$T_3$ space]].
We also have by definition of [[Definition:Normal Space|normal space]] that $T$ is a [[Definition:T1 Space|$T_1$ space]].
From [[T1 ... | Normal Space is Regular | https://proofwiki.org/wiki/Normal_Space_is_Regular | https://proofwiki.org/wiki/Normal_Space_is_Regular | [
"Regular Spaces",
"Normal Spaces"
] | [
"Definition:Normal Space",
"Definition:Regular Space"
] | [
"Definition:Normal Space",
"Normal Space is T3",
"Definition:T3 Space",
"Definition:Normal Space",
"Definition:T1 Space",
"T1 Space is T0",
"Definition:T0 Space",
"Definition:T3 Space",
"Definition:T0 Space",
"Definition:Regular Space"
] |
proofwiki-6425 | Product Space is T2 iff Factor Spaces are T2/Necessary Condition | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }$ be an indexed family of non-empty topological spaces for $\alpha$ in some indexing set $I$.
Let $\ds T = \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$.
Let each of $\struct {S_\alpha, \tau_\alpha}$ f... | Let each of $\struct {S_\alpha, \tau_\alpha}$ for $\alpha \in I$ be $T_2$ (Hausdorff) space.
Let $x, y \in S : x \ne y$.
Then $x_\alpha \ne y_\alpha$ for some $\alpha \in I$.
Since $\struct {S_\alpha, \tau_\alpha}$ is Hausdorff then:
:$\exists U, V \in \tau_\alpha: x_\alpha \in U, y_\alpha \in V : U \cap V = \O$
Let $\... | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological spaces]] for $\alpha$ in some [[Definition:Indexing Set|indexing set]] $I$.
Let $\ds T = \struct {S, \tau} = \prod_{\alpha \math... | Let each of $\struct {S_\alpha, \tau_\alpha}$ for $\alpha \in I$ be [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
Let $x, y \in S : x \ne y$.
Then $x_\alpha \ne y_\alpha$ for some $\alpha \in I$.
Since $\struct {S_\alpha, \tau_\alpha}$ is [[Definition:T2 Space|Hausdorff]] then:
:$\exists U, V \in \tau_\alpha: x_\... | Product Space is T2 iff Factor Spaces are T2/Necessary Condition | https://proofwiki.org/wiki/Product_Space_is_T2_iff_Factor_Spaces_are_T2/Necessary_Condition | https://proofwiki.org/wiki/Product_Space_is_T2_iff_Factor_Spaces_are_T2/Necessary_Condition | [
"Product Space is T2 iff Factor Spaces are T2"
] | [
"Definition:Indexing Set/Family",
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:T2 Space",
"Definition:T2 Space"
] | [
"Definition:T2 Space",
"Definition:T2 Space",
"Definition:Projection (Mapping Theory)",
"Preimage of Intersection under Mapping",
"Definition:Projection (Mapping Theory)",
"Definition:Preimage/Mapping",
"Definition:Product Topology",
"Definition:T2 Space"
] |
proofwiki-6426 | Product Space is T2 iff Factor Spaces are T2/Necessary Condition/Product of 2 Spaces | Let $T_\alpha = \struct {S_\alpha, \tau_\alpha}$ and $T_\beta = \struct {S_\beta, \tau_\beta}$ be topological spaces.
Let $T = T_\alpha \times T_\beta$ be the product space of $T_\alpha$ and $T_\beta$
Let $T_\alpha$ and $T_\beta$ both be $T_2$ (Hausdorff) space.
Then $T$ is also a $T_2$ (Hausdorff) space. | Let $T_\alpha$ and $T_\beta$ be $T_2$ (Hausdorff) space.
Let $\tuple {a, b}$ and $\tuple {c, d}$ be two distinct points of the product space $T$.
Let $a = c$.
Then as $\tuple {a, b} \ne \tuple {c, d}$ it follows that $b \ne d$.
As $T_\beta$ is Hausdorff, there exists two disjoint open sets $U, V \subseteq T_\beta$ such... | Let $T_\alpha = \struct {S_\alpha, \tau_\alpha}$ and $T_\beta = \struct {S_\beta, \tau_\beta}$ be [[Definition:Topological Space|topological spaces]].
Let $T = T_\alpha \times T_\beta$ be the [[Definition:Product Space (Topology) of Two Factor Spaces|product space]] of $T_\alpha$ and $T_\beta$
Let $T_\alpha$ and $T_\... | Let $T_\alpha$ and $T_\beta$ be [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
Let $\tuple {a, b}$ and $\tuple {c, d}$ be two [[Definition:Distinct|distinct]] points of the [[Definition:Product Space (Topology) of Two Factor Spaces|product space]] $T$.
Let $a = c$.
Then as $\tuple {a, b} \ne \tuple {c, d}$ it fol... | Product Space is T2 iff Factor Spaces are T2/Necessary Condition/Product of 2 Spaces | https://proofwiki.org/wiki/Product_Space_is_T2_iff_Factor_Spaces_are_T2/Necessary_Condition/Product_of_2_Spaces | https://proofwiki.org/wiki/Product_Space_is_T2_iff_Factor_Spaces_are_T2/Necessary_Condition/Product_of_2_Spaces | [
"Product Space is T2 iff Factor Spaces are T2"
] | [
"Definition:Topological Space",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:T2 Space",
"Definition:T2 Space"
] | [
"Definition:T2 Space",
"Definition:Distinct",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:T2 Space",
"Definition:Disjoint Sets",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Disjoint Sets",
"Definition:Product Space (Topology)/Two Factor Spaces... |
proofwiki-6427 | Domain of Continuous Injection to Hausdorff Space is Hausdorff | Let $T_\alpha = \struct {S_\alpha, \tau_\alpha}$ and $T_\beta = \struct {S_\beta, \tau_\beta}$ be topological spaces.
Let $f: S_\alpha \to S_\beta$ be a continuous mapping which is an injection.
If $T_\beta$ is a $T_2$ (Hausdorff) space, then $T_\alpha$ is also a $T_2$ (Hausdorff) space. | Let $x, y \in S_\alpha$ be distinct points.
We want to find disjoint open sets $U, V \in \tau_\alpha$ containing $x$ and $y$ respectively.
Since $f$ is injective the points $\map f x, \map f y \in S_\beta$ are distinct.
By assumption $T_\beta$ is Hausdorff.
Therefore we can choose disjoint open sets $U', V'$ of $T_\bet... | Let $T_\alpha = \struct {S_\alpha, \tau_\alpha}$ and $T_\beta = \struct {S_\beta, \tau_\beta}$ be [[Definition:Topological Space|topological spaces]].
Let $f: S_\alpha \to S_\beta$ be a [[Definition:Continuous Mapping (Topology)|continuous mapping]] which is an [[Definition:Injection|injection]].
If $T_\beta$ is a [... | Let $x, y \in S_\alpha$ be [[Definition:Distinct Elements|distinct]] [[Definition:Point|points]].
We want to find [[Definition:Disjoint Sets|disjoint]] [[Definition:Open Set (Topology)|open sets]] $U, V \in \tau_\alpha$ containing $x$ and $y$ respectively.
Since $f$ is [[Definition:Injection|injective]] the [[Defini... | Domain of Continuous Injection to Hausdorff Space is Hausdorff | https://proofwiki.org/wiki/Domain_of_Continuous_Injection_to_Hausdorff_Space_is_Hausdorff | https://proofwiki.org/wiki/Domain_of_Continuous_Injection_to_Hausdorff_Space_is_Hausdorff | [
"Hausdorff Spaces",
"Injections",
"Continuous Mappings"
] | [
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)",
"Definition:Injection",
"Definition:T2 Space",
"Definition:T2 Space"
] | [
"Definition:Distinct/Plural",
"Definition:Point",
"Definition:Disjoint Sets",
"Definition:Open Set/Topology",
"Definition:Injection",
"Definition:Point",
"Definition:Distinct/Plural",
"Definition:T2 Space",
"Definition:Disjoint Sets",
"Definition:Open Set/Topology",
"Definition:Continuous Mappin... |
proofwiki-6428 | Equivalence of Definitions of Homeomorphisms between Topological Spaces | Let $T_\alpha = \struct {S_\alpha, \tau_\alpha}$ and $T_\beta = \struct {S_\beta, \tau_\beta}$ be topological spaces.
Let $f: T_\alpha \to T_\beta$ be a bijection.
{{TFAE|def = Homeomorphism (Topological Spaces)|view = homeomorphism}} | === Definition 1 iff Definition 2 ===
Let $f$ and $f^{-1}$ both be continuous.
As $f$ is continuous, then by definition:
:$V \in \tau_\beta \implies f^{-1} \sqbrk V \in \tau_\alpha$
and as $f^{-1}$ is continuous, then by definition:
:$U \in \tau_\alpha \implies \paren {f^{-1} }^{-1} \sqbrk U = f \sqbrk U \in \tau_\beta... | Let $T_\alpha = \struct {S_\alpha, \tau_\alpha}$ and $T_\beta = \struct {S_\beta, \tau_\beta}$ be [[Definition:Topological Space|topological spaces]].
Let $f: T_\alpha \to T_\beta$ be a [[Definition:Bijection|bijection]].
{{TFAE|def = Homeomorphism (Topological Spaces)|view = homeomorphism}} | === Definition 1 iff Definition 2 ===
Let $f$ and $f^{-1}$ both be [[Definition:Continuous Mapping (Topology)|continuous]].
As $f$ is [[Definition:Continuous Mapping (Topology)|continuous]], then by definition:
:$V \in \tau_\beta \implies f^{-1} \sqbrk V \in \tau_\alpha$
and as $f^{-1}$ is [[Definition:Continuous Map... | Equivalence of Definitions of Homeomorphisms between Topological Spaces | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Homeomorphisms_between_Topological_Spaces | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Homeomorphisms_between_Topological_Spaces | [
"Homeomorphisms (Topological Spaces)"
] | [
"Definition:Topological Space",
"Definition:Bijection"
] | [
"Definition:Continuous Mapping (Topology)",
"Definition:Continuous Mapping (Topology)",
"Definition:Continuous Mapping (Topology)",
"Definition:Converse Statement",
"Definition:Continuous Mapping (Topology)",
"Definition:Continuous Mapping (Topology)",
"Definition:Continuous Mapping (Topology)",
"Defi... |
proofwiki-6429 | Quotient Space of Hausdorff Space is not necessarily Hausdorff | Let $T = \struct {S, \tau}$ be a $T_2$ (Hausdorff) space.
Let $\RR \subseteq S \times S$ be an equivalence relation on $S$.
Let $T_\RR := \struct {S / \RR, \tau_\RR}$ be the quotient space of $S$ by $\RR$.
Then $T_\RR$ is not necessarily also a $T_2$ (Hausdorff) space. | Consider the real number line with the Euclidean topology $\struct {\R, \tau}$.
By Real Number Line satisfies all Separation Axioms, $\struct {\R, \tau}$ is a $T_2$ (Hausdorff) space.
By Quotient Space of Real Line may not be T0, there is a relation $\RR$ on $\R$ such that the quotient space $\struct {\R / \RR, \tau_\R... | Let $T = \struct {S, \tau}$ be a [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
Let $\RR \subseteq S \times S$ be an [[Definition:Equivalence Relation|equivalence relation]] on $S$.
Let $T_\RR := \struct {S / \RR, \tau_\RR}$ be the [[Definition:Quotient Space (Topology)|quotient space]] of $S$ by $\RR$.
Then $T_\... | Consider the [[Definition:Real Number Line with Euclidean Topology|real number line with the Euclidean topology]] $\struct {\R, \tau}$.
By [[Real Number Line satisfies all Separation Axioms]], $\struct {\R, \tau}$ is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
By [[Quotient Space of Real Line may not be T0]], ... | Quotient Space of Hausdorff Space is not necessarily Hausdorff | https://proofwiki.org/wiki/Quotient_Space_of_Hausdorff_Space_is_not_necessarily_Hausdorff | https://proofwiki.org/wiki/Quotient_Space_of_Hausdorff_Space_is_not_necessarily_Hausdorff | [
"Quotient Spaces (Topology)",
"Hausdorff Spaces"
] | [
"Definition:T2 Space",
"Definition:Equivalence Relation",
"Definition:Quotient Topology/Quotient Space",
"Definition:T2 Space"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Real Number Line satisfies all Separation Axioms",
"Definition:T2 Space",
"Quotient Space of Real Line may not be T0",
"Definition:Quotient Topology/Quotient Space",
"Definition:T0 Space",
"Sequence of Implications of Separation Axioms"
... |
proofwiki-6430 | Existence and Uniqueness of Direct Limit of Sequence of Groups | Let $\sequence {G_n}_{n \mathop \in \N}$ be a sequence of groups.
Let $\sequence {g_n}_{n \mathop \in \N}: g_n: G_n \to G_{n + 1}$ be a sequence of group homomorphisms.
Then their direct limit $G_\infty$ exists and is unique up to unique isomorphism. | {{refactor|Transclude the lemmata|level = basic}} | Let $\sequence {G_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Group|groups]].
Let $\sequence {g_n}_{n \mathop \in \N}: g_n: G_n \to G_{n + 1}$ be a [[Definition:Sequence|sequence]] of [[Definition:Group Homomorphism|group homomorphisms]].
Then their [[Definition:Direct Limit of Seque... | {{refactor|Transclude the lemmata|level = basic}} | Existence and Uniqueness of Direct Limit of Sequence of Groups | https://proofwiki.org/wiki/Existence_and_Uniqueness_of_Direct_Limit_of_Sequence_of_Groups | https://proofwiki.org/wiki/Existence_and_Uniqueness_of_Direct_Limit_of_Sequence_of_Groups | [
"Group Theory",
"Existence and Uniqueness of Direct Limit of Sequence of Groups"
] | [
"Definition:Sequence",
"Definition:Group",
"Definition:Sequence",
"Definition:Group Homomorphism",
"Definition:Direct Limit of Sequence of Groups/Definition 1",
"Definition:Unique",
"Definition:Unique",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [] |
proofwiki-6431 | Topological Product of Compact Spaces/Finite Product | Let $T_1, T_2, \ldots, T_n$ be topological spaces.
Let $\ds \prod_{i \mathop = 1}^n T_i$ be the product space of $T_1, T_2, \ldots, T_n$.
Then $\ds \prod_{i \mathop = 1}^n T_i$ is compact {{iff}} all of $T_1, T_2, \ldots, T_n$ are compact. | Proof by induction:
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$\ds \prod_{i \mathop = 1}^n T_i$ is compact {{iff}} all of $T_1, T_2, \ldots, T_n$ are compact | Let $T_1, T_2, \ldots, T_n$ be [[Definition:Topological Space|topological spaces]].
Let $\ds \prod_{i \mathop = 1}^n T_i$ be the [[Definition:Product Space (Topology)|product space]] of $T_1, T_2, \ldots, T_n$.
Then $\ds \prod_{i \mathop = 1}^n T_i$ is [[Definition:Compact Topological Space|compact]] {{iff}} all of ... | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \prod_{i \mathop = 1}^n T_i$ is [[Definition:Compact Topological Space|compact]] {{iff}} all of $T_1, T_2, \ldots, T_n$ are [[Definition:Compact Topological Space|co... | Topological Product of Compact Spaces/Finite Product | https://proofwiki.org/wiki/Topological_Product_of_Compact_Spaces/Finite_Product | https://proofwiki.org/wiki/Topological_Product_of_Compact_Spaces/Finite_Product | [
"Topology"
] | [
"Definition:Topological Space",
"Definition:Product Space (Topology)",
"Definition:Compact Topological Space",
"Definition:Compact Topological Space"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Compact Topological Space",
"Definition:Compact Topological Space",
"Definition:Compact Topological Space",
"Definition:Compact Topological Space",
"Definition:Compact Topological Space",
"Definition:Compact Topological Space... |
proofwiki-6432 | Uniform Continuity on Metric Space does not imply Compactness | 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 uniformly continuous mapping on $A_1$.
Then it is not necessarily the case that $M_1$ is a compact metric space. | Let $M_1 = \struct {A_1, d_1}$ be any metric space which is not compact.
Let $I_{M_1}: M_1 \to M_1$ be the identity mapping.
From Identity Mapping is Uniformly Continuous, $I_{M_1}$ is uniformly continuous on $M_1$.
Hence the result.
{{qed}} | 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:Uniformly Continuous Mapping (Metric Spaces)|uniformly continuous mapping]] on $A_1$.
Then it is not necessarily the case that $M_1$ is a [[Definition:Compact Metric Spac... | Let $M_1 = \struct {A_1, d_1}$ be any [[Definition:Metric Space|metric space]] which is not [[Definition:Compact Metric Space|compact]].
Let $I_{M_1}: M_1 \to M_1$ be the [[Definition:Identity Mapping|identity mapping]].
From [[Identity Mapping is Uniformly Continuous]], $I_{M_1}$ is [[Definition:Uniformly Continuous... | Uniform Continuity on Metric Space does not imply Compactness | https://proofwiki.org/wiki/Uniform_Continuity_on_Metric_Space_does_not_imply_Compactness | https://proofwiki.org/wiki/Uniform_Continuity_on_Metric_Space_does_not_imply_Compactness | [
"Metric Spaces",
"Compact Metric Spaces",
"Uniformly Continuous Mappings"
] | [
"Definition:Metric Space",
"Definition:Uniform Continuity/Metric Space",
"Definition:Compact Space/Metric Space"
] | [
"Definition:Metric Space",
"Definition:Compact Space/Metric Space",
"Definition:Identity Mapping",
"Identity Mapping is Uniformly Continuous",
"Definition:Uniform Continuity/Metric Space"
] |
proofwiki-6433 | Uniformly Continuous Function is Continuous/Metric Space | Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_1, d_1}\right)$ be metric spaces.
Let the mapping $f: M_1 \to M_2$ be uniformly continuous on $M_1$.
Then $f$ is continuous on $M_1$. | Let $f$ be uniformly continuous on $M_1$.
Let $x \in M_1$.
Let $\epsilon > 0$.
As $f$ is uniformly continuous, $\exists \delta > 0$ such that:
:$\forall y \in M_1: d_1 \left({x, y}\right) < \delta: d_2 \left({f \left({x}\right), f \left({y}\right)}\right) < \epsilon$
Thus by definition $f$ is continuous at $x$.
{{qed}}... | Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_1, d_1}\right)$ be [[Definition:Metric Space|metric spaces]].
Let the [[Definition:Mapping|mapping]] $f: M_1 \to M_2$ be [[Definition:Uniformly Continuous Mapping (Metric Spaces)|uniformly continuous on $M_1$]].
Then $f$ is [[Definition:Continuous Mapping (Metr... | Let $f$ be [[Definition:Uniformly Continuous Mapping (Metric Spaces)|uniformly continuous on $M_1$]].
Let $x \in M_1$.
Let $\epsilon > 0$.
As $f$ is [[Definition:Uniformly Continuous Mapping (Metric Spaces)|uniformly continuous]], $\exists \delta > 0$ such that:
:$\forall y \in M_1: d_1 \left({x, y}\right) < \delta:... | Uniformly Continuous Function is Continuous/Metric Space | https://proofwiki.org/wiki/Uniformly_Continuous_Function_is_Continuous/Metric_Space | https://proofwiki.org/wiki/Uniformly_Continuous_Function_is_Continuous/Metric_Space | [
"Metric Spaces",
"Continuous Mappings on Metric Spaces",
"Uniformly Continuous Mappings"
] | [
"Definition:Metric Space",
"Definition:Mapping",
"Definition:Uniform Continuity/Metric Space",
"Definition:Continuous Mapping (Metric Space)"
] | [
"Definition:Uniform Continuity/Metric Space",
"Definition:Uniform Continuity/Metric Space",
"Definition:Continuous Mapping (Metric Space)",
"Category:Metric Spaces",
"Category:Continuous Mappings on Metric Spaces",
"Category:Uniformly Continuous Mappings"
] |
proofwiki-6434 | Uniformly Continuous Function is Continuous/Real Function | Let $I$ be an interval of $\R$.
Let $f: I \to \R$ be a uniformly continuous real function on $I$.
Then $f$ is continuous on $I$. | From Real Number Line is Metric Space, $\R$ under the Euclidean metric is a metric space.
The result follows by Uniformly Continuous Function is Continuous: Metric Space.
{{qed}} | Let $I$ be an [[Definition:Real Interval|interval]] of $\R$.
Let $f: I \to \R$ be a [[Definition:Uniformly Continuous Real Function|uniformly continuous real function]] on $I$.
Then $f$ is [[Definition:Continuous on Interval|continuous]] on $I$. | From [[Real Number Line is Metric Space]], $\R$ under the [[Definition:Euclidean Metric on Real Number Line|Euclidean metric]] is a [[Definition:Metric Space|metric space]].
The result follows by [[Uniformly Continuous Function is Continuous/Metric Space|Uniformly Continuous Function is Continuous: Metric Space]].
{{q... | Uniformly Continuous Function is Continuous/Real Function/Proof 1 | https://proofwiki.org/wiki/Uniformly_Continuous_Function_is_Continuous/Real_Function | https://proofwiki.org/wiki/Uniformly_Continuous_Function_is_Continuous/Real_Function/Proof_1 | [
"Uniformly Continuous Real Function is Continuous",
"Uniformly Continuous Real Functions",
"Continuous Real Functions",
"Real Analysis"
] | [
"Definition:Real Interval",
"Definition:Uniform Continuity/Real Function",
"Definition:Continuous Real Function/Interval"
] | [
"Real Number Line is Metric Space",
"Definition:Euclidean Metric/Real Number Line",
"Definition:Metric Space",
"Uniformly Continuous Function is Continuous/Metric Space"
] |
proofwiki-6435 | Uniformly Continuous Function is Continuous/Real Function | Let $I$ be an interval of $\R$.
Let $f: I \to \R$ be a uniformly continuous real function on $I$.
Then $f$ is continuous on $I$. | Let $x \in I$.
Let $\epsilon \in \R_{>0}$.
As $f$ is uniformly continuous:
:$\exists \delta \in \R_{>0}: \paren {x, y \in I, \size {x - y} < \delta \implies \size {\map f x - \map f y} < \epsilon}$
Then, for all $y \in I$ such that $\size {x - y} < \delta$:
:$\size {\map f x - \map f y} < \epsilon$
Thus by definition ... | Let $I$ be an [[Definition:Real Interval|interval]] of $\R$.
Let $f: I \to \R$ be a [[Definition:Uniformly Continuous Real Function|uniformly continuous real function]] on $I$.
Then $f$ is [[Definition:Continuous on Interval|continuous]] on $I$. | Let $x \in I$.
Let $\epsilon \in \R_{>0}$.
As $f$ is [[Definition:Uniformly Continuous Real Function|uniformly continuous]]:
:$\exists \delta \in \R_{>0}: \paren {x, y \in I, \size {x - y} < \delta \implies \size {\map f x - \map f y} < \epsilon}$
Then, for all $y \in I$ such that $\size {x - y} < \delta$:
:$\size... | Uniformly Continuous Function is Continuous/Real Function/Proof 2 | https://proofwiki.org/wiki/Uniformly_Continuous_Function_is_Continuous/Real_Function | https://proofwiki.org/wiki/Uniformly_Continuous_Function_is_Continuous/Real_Function/Proof_2 | [
"Uniformly Continuous Real Function is Continuous",
"Uniformly Continuous Real Functions",
"Continuous Real Functions",
"Real Analysis"
] | [
"Definition:Real Interval",
"Definition:Uniform Continuity/Real Function",
"Definition:Continuous Real Function/Interval"
] | [
"Definition:Uniform Continuity/Real Function",
"Definition:Continuous Real Function/Point",
"Definition:Continuous Real Function/Interval"
] |
proofwiki-6436 | Young's Inequality for Increasing Functions/Equality | Let $a_0$ and $b_0$ be strictly positive real numbers.
Let $f: \closedint 0 {a_0} \to \closedint 0 {b_0}$ be a strictly increasing bijection.
Let $a$ and $b$ be real numbers such that $0 \le a \le a_0$ and $0 \le b \le b_0$.
Then $b = \map f a$ {{iff}}:
:$\ds a b = \int_0^a \map f u \rd u + \int_0^b \map {f^{-1} } v \r... | === Necessary Condition ===
{{ProofWanted}} | Let $a_0$ and $b_0$ be [[Definition:Strictly Positive Real Number|strictly positive real numbers]].
Let $f: \closedint 0 {a_0} \to \closedint 0 {b_0}$ be a [[Definition:Strictly Increasing Real Function|strictly increasing]] [[Definition:Bijection|bijection]].
Let $a$ and $b$ be [[Definition:Real Number|real numbers]... | === Necessary Condition ===
{{ProofWanted}} | Young's Inequality for Increasing Functions/Equality | https://proofwiki.org/wiki/Young's_Inequality_for_Increasing_Functions/Equality | https://proofwiki.org/wiki/Young's_Inequality_for_Increasing_Functions/Equality | [
"Young's Inequality for Increasing Functions"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Strictly Increasing/Real Function",
"Definition:Bijection",
"Definition:Real Number",
"Definition:Definite Integral/Darboux"
] | [] |
proofwiki-6437 | Category of Finite Sets is Cartesian Closed | Let $\mathbf{Finset}$ be the category of finite sets.
Then $\mathbf{Finset}$ is Cartesian closed. | {{MissingLinks}}
{{Recall|Cartesian Closed Category}}
{{:Definition:Cartesian Closed Category}}
Recall Cartesian Product is Set Product.
By Cardinality of Cartesian Product of Finite Sets, the finite product of sets has finite cardinality.
Hence $\mathbf{C}$ has finite products.
By Cardinality of Set of All Mappings, t... | Let $\mathbf{Finset}$ be the [[Definition:Category of Finite Sets|category of finite sets]].
Then $\mathbf{Finset}$ is [[Definition:Cartesian Closed Category|Cartesian closed]]. | {{MissingLinks}}
{{Recall|Cartesian Closed Category}}
{{:Definition:Cartesian Closed Category}}
Recall [[Cartesian Product is Set Product]].
By [[Cardinality of Cartesian Product of Finite Sets/General Result|Cardinality of Cartesian Product of Finite Sets]], the finite product of sets has finite cardinality.
Hence... | Category of Finite Sets is Cartesian Closed | https://proofwiki.org/wiki/Category_of_Finite_Sets_is_Cartesian_Closed | https://proofwiki.org/wiki/Category_of_Finite_Sets_is_Cartesian_Closed | [
"Category of Finite Sets"
] | [
"Definition:Category of Finite Sets",
"Definition:Cartesian Closed Category"
] | [
"Cartesian Product is Set Product",
"Cardinality of Cartesian Product of Finite Sets/General Result",
"Cardinality of Set of All Mappings",
"Category of Sets is Cartesian Closed"
] |
proofwiki-6438 | Components of Separation are Clopen | Let $T = \struct {S, \tau}$ be a topological space.
Let $A \mid B$ be a separation of $T$.
Then both $A$ and $B$ are clopen in $T$. | From Set with Relative Complement forms Partition:
:$A = \relcomp S B$
and:
:$B = \relcomp S A$
where $\complement_S$ denotes the complement relative to $S$.
As $A$ and $B$ are both open, it follows by definition that $A$ and $B$ are also both closed.
That is, by definition, they are clopen.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A \mid B$ be a [[Definition:Separation (Topology)|separation]] of $T$.
Then both $A$ and $B$ are [[Definition:Clopen Set|clopen]] in $T$. | From [[Set with Relative Complement forms Partition]]:
:$A = \relcomp S B$
and:
:$B = \relcomp S A$
where $\complement_S$ denotes the [[Definition:Relative Complement|complement relative to $S$]].
As $A$ and $B$ are both [[Definition:Open Set (Topology)|open]], it follows by definition that $A$ and $B$ are also both [... | Components of Separation are Clopen | https://proofwiki.org/wiki/Components_of_Separation_are_Clopen | https://proofwiki.org/wiki/Components_of_Separation_are_Clopen | [
"Disconnected Spaces"
] | [
"Definition:Topological Space",
"Definition:Separation (Topology)",
"Definition:Clopen Set"
] | [
"Set Difference and Intersection form Partition/Corollary 2",
"Definition:Relative Complement",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Clopen Set"
] |
proofwiki-6439 | Equivalence of Definitions of Connected Topological Space/No Separation iff No Union of Closed Sets | {{TFAE|def = Connected Topological Space}}
Let $T = \struct {S, \tau}$ be a topological space. | From Biconditional Equivalent to Biconditional of Negations it follows that the statement can be expressed as:
:$T$ admits a separation
{{iff}}:
:there exist two closed sets of $T$ which form a (set) partition of $S$.
By definition, a separation of $T$ is a (set) partition of $S$ by $A, B$ which are open in $T$.
From C... | {{TFAE|def = Connected Topological Space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]]. | From [[Biconditional Equivalent to Biconditional of Negations/Formulation 1|Biconditional Equivalent to Biconditional of Negations]] it follows that the statement can be expressed as:
:$T$ admits a [[Definition:Separation (Topology)|separation]]
{{iff}}:
:there exist two [[Definition:Closed Set (Topology)|closed sets]... | Equivalence of Definitions of Connected Topological Space/No Separation iff No Union of Closed Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Connected_Topological_Space/No_Separation_iff_No_Union_of_Closed_Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Connected_Topological_Space/No_Separation_iff_No_Union_of_Closed_Sets | [
"Equivalence of Definitions of Connected Topological Space"
] | [
"Definition:Topological Space"
] | [
"Biconditional Equivalent to Biconditional of Negations/Formulation 1",
"Definition:Separation (Topology)",
"Definition:Closed Set/Topology",
"Definition:Set Partition",
"Definition:Separation (Topology)",
"Definition:Set Partition",
"Definition:Open Set/Topology",
"Complements of Components of Two-Co... |
proofwiki-6440 | Empty Set Satisfies Topology Axioms | Let $T = \struct {\O, \set \O}$ where $\O$ denotes the empty set.
Then $T$ satisfies the open set axioms for a topological space. | We proceed to verify the open set axioms for $\set \O$ to be a topology on $\O$.
Let $\tau = \set \O$. | Let $T = \struct {\O, \set \O}$ where $\O$ denotes the [[Definition:Empty Set|empty set]].
Then $T$ satisfies the [[Axiom:Open Set Axioms|open set axioms]] for a [[Definition:Topological Space|topological space]]. | We proceed to verify the [[Axiom:Open Set Axioms|open set axioms]] for $\set \O$ to be a [[Definition:Topology|topology]] on $\O$.
Let $\tau = \set \O$. | Empty Set Satisfies Topology Axioms | https://proofwiki.org/wiki/Empty_Set_Satisfies_Topology_Axioms | https://proofwiki.org/wiki/Empty_Set_Satisfies_Topology_Axioms | [
"Topology",
"Empty Set",
"Empty Topological Space"
] | [
"Definition:Empty Set",
"Axiom:Open Set Axioms",
"Definition:Topological Space"
] | [
"Axiom:Open Set Axioms",
"Definition:Topology",
"Axiom:Open Set Axioms"
] |
proofwiki-6441 | Complements of Components of Two-Component Partition form Partition | Let $S$ be a set with at least two elements.
Let $A, B \subseteq S$.
Let $\complement_S$ denote the complement relative to $S$.
$A \mid B$ is a partition of $S$ {{iff}} $\relcomp S A \mid \relcomp S B$ is a partition of $S$. | === Necessary Condition ===
Let $A \mid B$ be a partition of $S$.
That is, by definition:
{{begin-eqn}}
{{eqn | n = 1
| l = A \cap B
| r = \O
}}
{{eqn | n = 2
| l = A \cup B
| r = S
}}
{{eqn | n = 3
| l = A, B
| o = \ne
| r = \O
}}
{{end-eqn}}
Thus:
{{begin-eqn}}
{{eqn | n = 1
... | Let $S$ be a [[Definition:Set|set]] with at least two [[Definition:Element|elements]].
Let $A, B \subseteq S$.
Let $\complement_S$ denote the [[Definition:Relative Complement|complement relative to $S$]].
$A \mid B$ is a [[Definition:Set Partition|partition]] of $S$ {{iff}} $\relcomp S A \mid \relcomp S B$ is a [[D... | === Necessary Condition ===
Let $A \mid B$ be a [[Definition:Set Partition|partition]] of $S$.
That is, by definition:
{{begin-eqn}}
{{eqn | n = 1
| l = A \cap B
| r = \O
}}
{{eqn | n = 2
| l = A \cup B
| r = S
}}
{{eqn | n = 3
| l = A, B
| o = \ne
| r = \O
}}
{{end-eqn}}
T... | Complements of Components of Two-Component Partition form Partition | https://proofwiki.org/wiki/Complements_of_Components_of_Two-Component_Partition_form_Partition | https://proofwiki.org/wiki/Complements_of_Components_of_Two-Component_Partition_form_Partition | [
"Relative Complement",
"Set Partitions"
] | [
"Definition:Set",
"Definition:Element",
"Definition:Relative Complement",
"Definition:Set Partition",
"Definition:Set Partition"
] | [
"Definition:Set Partition",
"Relative Complement of Empty Set",
"De Morgan's Laws (Set Theory)/Relative Complement/Complement of Intersection",
"Relative Complement with Self is Empty Set",
"De Morgan's Laws (Set Theory)/Relative Complement/Complement of Union",
"Relative Complement of Empty Set",
"Rule... |
proofwiki-6442 | Sigma-Algebra Closed under Finite Intersection | Let $A_1, \ldots, A_n \in \Sigma$.
Then $\ds \bigcap_{k \mathop = 1}^n A_k \in \Sigma$. | Define for $k \in \N, k > n: A_k = X$.
By axiom $(1)$ of a $\sigma$-algebra, it follows that $\forall k \in \N, k > n: A_k \in \Sigma$.
From Sigma-Algebra Closed under Countable Intersection, it follows that $\ds \bigcap_{k \mathop \in \N} A_k = \bigcap_{k \mathop = 1}^n A_k \in \Sigma$.
{{qed}}
Category:Sigma-Algebras... | Let $A_1, \ldots, A_n \in \Sigma$.
Then $\ds \bigcap_{k \mathop = 1}^n A_k \in \Sigma$. | Define for $k \in \N, k > n: A_k = X$.
By axiom $(1)$ of a [[Definition:Sigma-Algebra|$\sigma$-algebra]], it follows that $\forall k \in \N, k > n: A_k \in \Sigma$.
From [[Sigma-Algebra Closed under Countable Intersection]], it follows that $\ds \bigcap_{k \mathop \in \N} A_k = \bigcap_{k \mathop = 1}^n A_k \in \Sigm... | Sigma-Algebra Closed under Finite Intersection | https://proofwiki.org/wiki/Sigma-Algebra_Closed_under_Finite_Intersection | https://proofwiki.org/wiki/Sigma-Algebra_Closed_under_Finite_Intersection | [
"Sigma-Algebras"
] | [] | [
"Definition:Sigma-Algebra",
"Sigma-Algebra Closed under Countable Intersection",
"Category:Sigma-Algebras"
] |
proofwiki-6443 | Equivalence of Definitions of Connected Topological Space/No Union of Closed Sets implies No Subsets with Empty Boundary | Let $T = \struct {S, \tau}$ be a topological space.
Let $T$ have no two disjoint non-empty closed sets whose union is $S$.
Then the only subsets of $S$ whose boundary is empty are $S$ and $\O$. | Let $H \subseteq S$ be a non-empty subset whose boundary $\partial H$ is empty.
Thus:
{{begin-eqn}}
{{eqn | l = \partial H
| r = \O
| c = {{hypothesis}}
}}
{{eqn | ll= \leadsto
| l = H^- \cap \paren {S \setminus H}^-
| r = \O
| c = Boundary is Intersection of Closure with Closure of Comple... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T$ have no two [[Definition:Disjoint Sets|disjoint]] [[Definition:Non-Empty Set|non-empty]] [[Definition:Closed Set (Topology)|closed sets]] whose [[Definition:Set Union|union]] is $S$.
Then the only [[Definition:Subset|subsets... | Let $H \subseteq S$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] whose [[Definition:Boundary (Topology)|boundary]] $\partial H$ is [[Definition:Empty Set|empty]].
Thus:
{{begin-eqn}}
{{eqn | l = \partial H
| r = \O
| c = {{hypothesis}}
}}
{{eqn | ll= \leadsto
| l = H^- \c... | Equivalence of Definitions of Connected Topological Space/No Union of Closed Sets implies No Subsets with Empty Boundary | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Connected_Topological_Space/No_Union_of_Closed_Sets_implies_No_Subsets_with_Empty_Boundary | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Connected_Topological_Space/No_Union_of_Closed_Sets_implies_No_Subsets_with_Empty_Boundary | [
"Equivalence of Definitions of Connected Topological Space"
] | [
"Definition:Topological Space",
"Definition:Disjoint Sets",
"Definition:Non-Empty Set",
"Definition:Closed Set/Topology",
"Definition:Set Union",
"Definition:Subset",
"Definition:Boundary (Topology)",
"Definition:Empty Set"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Boundary (Topology)",
"Definition:Empty Set",
"Boundary is Intersection of Closure with Closure of Complement",
"Topological Closure is Closed",
"Definition:Closed Set/Topology",
"Union of Closure with Closure of Complement is Whole Space",
... |
proofwiki-6444 | Equivalence of Definitions of Connected Topological Space/No Subsets with Empty Boundary implies No Clopen Sets | Let $T = \struct {S, \tau}$ be a topological space.
Let $T$ be such that the only subsets of $S$ whose boundary is empty are $S$ and $\O$.
Then the only clopen sets of $T$ are $S$ and $\O$. | Let $H \subseteq S$ be a clopen set of $T$.
From Set is Clopen iff Boundary is Empty, $H$ has an empty boundary.
We have {{hypothesis}} that $H = S$ or $H = \O$.
That is, the only clopen sets of $T$ are $S$ and $\O$. | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T$ be such that the only [[Definition:Subset|subsets]] of $S$ whose [[Definition:Boundary (Topology)|boundary]] is [[Definition:Empty Set|empty]] are $S$ and $\O$.
Then the only [[Definition:Clopen Set|clopen sets]] of $T$ are ... | Let $H \subseteq S$ be a [[Definition:Clopen Set|clopen set]] of $T$.
From [[Set is Clopen iff Boundary is Empty]], $H$ has an [[Definition:Empty Set|empty]] [[Definition:Boundary (Topology)|boundary]].
We have {{hypothesis}} that $H = S$ or $H = \O$.
That is, the only [[Definition:Clopen Set|clopen sets]] of $T$ ar... | Equivalence of Definitions of Connected Topological Space/No Subsets with Empty Boundary implies No Clopen Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Connected_Topological_Space/No_Subsets_with_Empty_Boundary_implies_No_Clopen_Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Connected_Topological_Space/No_Subsets_with_Empty_Boundary_implies_No_Clopen_Sets | [
"Equivalence of Definitions of Connected Topological Space"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Boundary (Topology)",
"Definition:Empty Set",
"Definition:Clopen Set"
] | [
"Definition:Clopen Set",
"Set is Clopen iff Boundary is Empty",
"Definition:Empty Set",
"Definition:Boundary (Topology)",
"Definition:Clopen Set"
] |
proofwiki-6445 | Equivalence of Definitions of Connected Topological Space/No Clopen Sets implies No Union of Separated Sets | Let $T = \struct {S, \tau}$ be a topological space.
Let the only clopen sets of $T$ be $S$ and $\O$.
Then there are no two non-empty separated sets of $T$ whose union is $S$. | Suppose $A$ and $B$ are separated subsets of $T$ such that $A \cup B = S$.
By definition of separated sets:
:$A \cap B^- = \O$
Then:
{{begin-eqn}}
{{eqn | l = S
| r = A \cup B
| c =
}}
{{eqn | o = \subseteq
| r = A \cup B^-
| c = Set is Subset of its Topological Closure
}}
{{eqn | o = \subseteq... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let the only [[Definition:Clopen Set|clopen sets]] of $T$ be $S$ and $\O$.
Then there are no two [[Definition:Non-Empty Set|non-empty]] [[Definition:Separated Sets|separated sets]] of $T$ whose [[Definition:Set Union|union]] is $S$. | Suppose $A$ and $B$ are [[Definition:Separated Sets|separated subsets]] of $T$ such that $A \cup B = S$.
By definition of [[Definition:Separated Sets|separated sets]]:
:$A \cap B^- = \O$
Then:
{{begin-eqn}}
{{eqn | l = S
| r = A \cup B
| c =
}}
{{eqn | o = \subseteq
| r = A \cup B^-
| c = ... | Equivalence of Definitions of Connected Topological Space/No Clopen Sets implies No Union of Separated Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Connected_Topological_Space/No_Clopen_Sets_implies_No_Union_of_Separated_Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Connected_Topological_Space/No_Clopen_Sets_implies_No_Union_of_Separated_Sets | [
"Equivalence of Definitions of Connected Topological Space"
] | [
"Definition:Topological Space",
"Definition:Clopen Set",
"Definition:Non-Empty Set",
"Definition:Separated Sets",
"Definition:Set Union"
] | [
"Definition:Separated Sets",
"Definition:Separated Sets",
"Set is Subset of its Topological Closure",
"Topological Closure is Closed",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Separated Sets",
"Definition:Open Set/Topology",
"Definition:Clopen Set",
"Definition... |
proofwiki-6446 | Equivalence of Definitions of Connected Topological Space/No Union of Separated Sets implies No Continuous Surjection to Discrete Two-Point Space | Let $T = \struct {S, \tau}$ be a topological space.
Let $T$ be such that there are no two non-empty separated sets whose union is $S$.
Then there exists no continuous surjection from $T$ onto a discrete two-point space. | Let $T = \struct {S, \tau}$ be a topological space such that there are no two non-empty separated sets whose union is $S$.
Let $D = \struct {\set {0, 1}, \tau}$ be the discrete two-point space on $\set {0, 1}$.
{{AimForCont}} $f: T \to \set {0, 1}$ is a continuous surjection.
By definition of continuous mapping:
:$\map... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T$ be such that there are no two [[Definition:Non-Empty Set|non-empty]] [[Definition:Separated Sets|separated sets]] whose [[Definition:Set Union|union]] is $S$.
Then there exists no [[Definition:Everywhere Continuous Mapping (... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] such that there are no two [[Definition:Non-Empty Set|non-empty]] [[Definition:Separated Sets|separated sets]] whose [[Definition:Set Union|union]] is $S$.
Let $D = \struct {\set {0, 1}, \tau}$ be the [[Definition:Discrete Space|discre... | Equivalence of Definitions of Connected Topological Space/No Union of Separated Sets implies No Continuous Surjection to Discrete Two-Point Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Connected_Topological_Space/No_Union_of_Separated_Sets_implies_No_Continuous_Surjection_to_Discrete_Two-Point_Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Connected_Topological_Space/No_Union_of_Separated_Sets_implies_No_Continuous_Surjection_to_Discrete_Two-Point_Space | [
"Equivalence of Definitions of Connected Topological Space"
] | [
"Definition:Topological Space",
"Definition:Non-Empty Set",
"Definition:Separated Sets",
"Definition:Set Union",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Surjection",
"Definition:Discrete Topology"
] | [
"Definition:Topological Space",
"Definition:Non-Empty Set",
"Definition:Separated Sets",
"Definition:Set Union",
"Definition:Discrete Topology",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Surjection",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Open Set... |
proofwiki-6447 | Equivalence of Definitions of Connected Topological Space/No Continuous Surjection to Discrete Two-Point Space implies No Separation | Let $T = \struct {S, \tau}$ be a topological space.
Let $T$ be such that there exists no continuous surjection from $T$ onto a discrete two-point space.
Then there exist no open sets $A, B \in \tau$ such that $A, B \ne \O$, $A \cup B = S$ and $A \cap B = \O$. | Let $T = \struct {S, \tau}$ be a topological space such that there exists no continuous surjection from $T$ onto a discrete two-point space.
Let $D = \struct {\set {0, 1}, \tau}$ be the discrete two-point space on $\left\{{0, 1}\right\}$.
Let $A$ and $B$ be disjoint open sets of $T$ such that $A \cup B = S$.
The aim is... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T$ be such that there exists no [[Definition:Everywhere Continuous Mapping (Topology)|continuous]] [[Definition:Surjection|surjection]] from $T$ onto a [[Definition:Discrete Topology|discrete two-point space]].
Then there exist... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] such that there exists no [[Definition:Everywhere Continuous Mapping (Topology)|continuous]] [[Definition:Surjection|surjection]] from $T$ onto a [[Definition:Discrete Topology|discrete two-point space]].
Let $D = \struct {\set {0, 1}... | Equivalence of Definitions of Connected Topological Space/No Continuous Surjection to Discrete Two-Point Space implies No Separation | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Connected_Topological_Space/No_Continuous_Surjection_to_Discrete_Two-Point_Space_implies_No_Separation | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Connected_Topological_Space/No_Continuous_Surjection_to_Discrete_Two-Point_Space_implies_No_Separation | [
"Equivalence of Definitions of Connected Topological Space"
] | [
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Surjection",
"Definition:Discrete Topology",
"Definition:Open Set/Topology"
] | [
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Surjection",
"Definition:Discrete Topology",
"Definition:Discrete Topology",
"Definition:Disjoint Sets",
"Definition:Open Set/Topology",
"Definition:Empty Set",
"Definition:Mapping",
"Definition:Prei... |
proofwiki-6448 | Existence of Vector Space Bases implies Axiom of Choice | The supposition that every vector space has a basis, along with the Zermelo-Fraenkel axioms, implies that the {{axiom-link|Choice}} holds. | [http://www.math.lsa.umich.edu/~ablass/bases-AC.pdf Blass, 1984]
{{proof wanted}} | The supposition that every [[Definition:Vector Space|vector space]] has a [[Definition:Basis of Vector Space|basis]], along with the [[Axiom:Zermelo-Fraenkel Axioms|Zermelo-Fraenkel axioms]], implies that the {{axiom-link|Choice}} holds. | [http://www.math.lsa.umich.edu/~ablass/bases-AC.pdf Blass, 1984]
{{proof wanted}} | Existence of Vector Space Bases implies Axiom of Choice | https://proofwiki.org/wiki/Existence_of_Vector_Space_Bases_implies_Axiom_of_Choice | https://proofwiki.org/wiki/Existence_of_Vector_Space_Bases_implies_Axiom_of_Choice | [
"Bases of Vector Spaces",
"Axiom of Choice"
] | [
"Definition:Vector Space",
"Definition:Basis of Vector Space",
"Axiom:Zermelo-Fraenkel Axioms"
] | [] |
proofwiki-6449 | Deterministic Time Hierarchy Theorem | Let $\map f n$ be a time-constructible function.
Then there exists a decision problem which:
:can be solved in worst-case deterministic time $\map f {2 n + 1}^3$
but:
:cannot be solved in worst-case deterministic time $\map f n$.
In other words, the complexity class $\map {\mathsf {DTIME} } {\map f n} \subsetneq \map {... | Let $H_f$ be a set defined as follows:
:$H_f = \set {\tuple {\sqbrk M, x}: \text {$M$ accepts $x$ in $\map f {\size x}$ steps} }$
where:
:$M$ is a (deterministic) Turing machine
:$x$ is its input (the initial contents of its tape)
:$\sqbrk M$ denotes an input that encodes the Turing machine $M$
Let $m$ be the size of $... | Let $\map f n$ be a [[Definition:Time-Constructible Function|time-constructible function]].
Then there exists a [[Definition:Decision Problem|decision problem]] which:
:can be solved in [[Definition:Worst-Case Deterministic Time|worst-case deterministic time]] $\map f {2 n + 1}^3$
but:
:cannot be solved in [[Definitio... | Let $H_f$ be a [[Definition:Set|set]] defined as follows:
:$H_f = \set {\tuple {\sqbrk M, x}: \text {$M$ accepts $x$ in $\map f {\size x}$ steps} }$
where:
:$M$ is a [[Definition:Turing Machine|(deterministic) Turing machine]]
:$x$ is its input (the initial contents of its tape)
:$\sqbrk M$ denotes an input that enco... | Deterministic Time Hierarchy Theorem | https://proofwiki.org/wiki/Deterministic_Time_Hierarchy_Theorem | https://proofwiki.org/wiki/Deterministic_Time_Hierarchy_Theorem | [
"Complexity Theory",
"Computer Science",
"Named Theorems"
] | [
"Definition:Time-Constructible Function",
"Definition:Decision Problem",
"Definition:Worst-Case Deterministic Time",
"Definition:Worst-Case Deterministic Time",
"Definition:Complexity Class"
] | [
"Definition:Set",
"Definition:Turing Machine",
"Definition:Turing Machine",
"Definition:Turing Machine",
"Definition:Turing Machine",
"Definition:Turing Machine",
"Definition:Turing Machine",
"Definition:Contradiction",
"Definition:Contradiction",
"Proof by Contradiction",
"Category:Complexity T... |
proofwiki-6450 | Exponentiation Functor is Functor | Let $\mathbf C$ be a Cartesian closed metacategory.
Let $A$ be an object of $\mathbf C$.
Let $\left({-}\right)^A: \mathbf C \to \mathbf C$ be the exponentiation functor.
Then $\left({-}\right)^A$ is a functor. | Let $B$ be an object of $\mathbf C$.
Let $\epsilon_B: B^A \times A \to B$ be the evaluation morphism at $B$.
Then, since:
{{begin-eqn}}
{{eqn | l = \operatorname{id}_B \epsilon_B
| r = \epsilon_B
}}
{{eqn | r = \epsilon_B \operatorname{id}_{B^A \times A}
}}
{{eqn | r = \map {\epsilon_B} {\operatorname{id}_{B^A} \... | Let $\mathbf C$ be a [[Definition:Cartesian Closed Category|Cartesian closed metacategory]].
Let $A$ be an [[Definition:Object (Category Theory)|object]] of $\mathbf C$.
Let $\left({-}\right)^A: \mathbf C \to \mathbf C$ be the [[Definition:Exponentiation Functor|exponentiation functor]].
Then $\left({-}\right)^A$ i... | Let $B$ be an [[Definition:Object (Category Theory)|object]] of $\mathbf C$.
Let $\epsilon_B: B^A \times A \to B$ be the [[Definition:Evaluation Morphism|evaluation morphism]] at $B$.
Then, since:
{{begin-eqn}}
{{eqn | l = \operatorname{id}_B \epsilon_B
| r = \epsilon_B
}}
{{eqn | r = \epsilon_B \operatorname{i... | Exponentiation Functor is Functor | https://proofwiki.org/wiki/Exponentiation_Functor_is_Functor | https://proofwiki.org/wiki/Exponentiation_Functor_is_Functor | [
"Functors"
] | [
"Definition:Cartesian Closed Category",
"Definition:Object (Category Theory)",
"Definition:Exponentiation Functor",
"Definition:Functor/Covariant"
] | [
"Definition:Object (Category Theory)",
"Definition:Exponential (Category Theory)/Evaluation",
"Definition:Composable Morphisms",
"Definition:Morphism",
"Definition:Exponential UMP",
"Definition:Functor/Covariant"
] |
proofwiki-6451 | Euler's Cosine Identity | :$\cos z = \dfrac {e^{i z} + e^{-i z} } 2$ | Recall the definition of the cosine function:
{{begin-eqn}}
{{eqn | l = \cos z
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | r = 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - \frac {z^6} {6!} + \cdots + \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!} + \cdots
... | :$\cos z = \dfrac {e^{i z} + e^{-i z} } 2$ | Recall the definition of the [[Definition:Complex Cosine Function|cosine function]]:
{{begin-eqn}}
{{eqn | l = \cos z
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | r = 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - \frac {z^6} {6!} + \cdots + \paren {-1}^n \f... | Euler's Cosine Identity/Proof 1 | https://proofwiki.org/wiki/Euler's_Cosine_Identity | https://proofwiki.org/wiki/Euler's_Cosine_Identity/Proof_1 | [
"Euler's Identities",
"Euler's Cosine Identity",
"Cosine Function"
] | [] | [
"Definition:Cosine/Complex Function",
"Definition:Exponential Function/Complex/Power Series Expansion",
"Cosine Function is Absolutely Convergent",
"Definition:Even Integer",
"Definition:Odd Integer"
] |
proofwiki-6452 | Euler's Cosine Identity | :$\cos z = \dfrac {e^{i z} + e^{-i z} } 2$ | Recall Euler's Formula:
:$e^{i z} = \cos z + i \sin z$
Then, starting from the {{RHS}}:
{{begin-eqn}}
{{eqn | l = \frac {e^{i z} + e^{-i z} } 2
| r = \frac {\cos z + i \sin z + \map \cos {-z} + i \map \sin {-z} } 2
}}
{{eqn | r = \frac {\cos z + \map \cos {-z} } 2
| c = Sine Function is Odd
}}
{{eqn | r = \... | :$\cos z = \dfrac {e^{i z} + e^{-i z} } 2$ | Recall [[Euler's Formula]]:
:$e^{i z} = \cos z + i \sin z$
Then, starting from the {{RHS}}:
{{begin-eqn}}
{{eqn | l = \frac {e^{i z} + e^{-i z} } 2
| r = \frac {\cos z + i \sin z + \map \cos {-z} + i \map \sin {-z} } 2
}}
{{eqn | r = \frac {\cos z + \map \cos {-z} } 2
| c = [[Sine Function is Odd]]
}}
{{... | Euler's Cosine Identity/Proof 2 | https://proofwiki.org/wiki/Euler's_Cosine_Identity | https://proofwiki.org/wiki/Euler's_Cosine_Identity/Proof_2 | [
"Euler's Identities",
"Euler's Cosine Identity",
"Cosine Function"
] | [] | [
"Euler's Formula",
"Sine Function is Odd",
"Cosine Function is Even"
] |
proofwiki-6453 | Euler's Cosine Identity | :$\cos z = \dfrac {e^{i z} + e^{-i z} } 2$ | {{begin-eqn}}
{{eqn | n = 1
| l = e^{i z}
| r = \cos z + i \sin z
| c = Euler's Formula
}}
{{eqn | n = 2
| l = e^{-i z}
| r = \cos z - i \sin z
| c = {{Corollary|Euler's Formula}}
}}
{{eqn | ll= \leadsto
| l = e^{i z} + e^{-i z}
| r = \paren {\cos z + i \sin z} + \paren {... | :$\cos z = \dfrac {e^{i z} + e^{-i z} } 2$ | {{begin-eqn}}
{{eqn | n = 1
| l = e^{i z}
| r = \cos z + i \sin z
| c = [[Euler's Formula]]
}}
{{eqn | n = 2
| l = e^{-i z}
| r = \cos z - i \sin z
| c = {{Corollary|Euler's Formula}}
}}
{{eqn | ll= \leadsto
| l = e^{i z} + e^{-i z}
| r = \paren {\cos z + i \sin z} + \par... | Euler's Cosine Identity/Proof 3 | https://proofwiki.org/wiki/Euler's_Cosine_Identity | https://proofwiki.org/wiki/Euler's_Cosine_Identity/Proof_3 | [
"Euler's Identities",
"Euler's Cosine Identity",
"Cosine Function"
] | [] | [
"Euler's Formula"
] |
proofwiki-6454 | Euler's Cosine Identity | :$\cos z = \dfrac {e^{i z} + e^{-i z} } 2$ | Recall the definition of the real cosine function:
{{begin-eqn}}
{{eqn | l = \cos x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n!} }
| c =
}}
{{eqn | r = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \frac {x^6} {6!} + \cdots + \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} + \... | :$\cos z = \dfrac {e^{i z} + e^{-i z} } 2$ | Recall the definition of the [[Definition:Real Cosine Function|real cosine function]]:
{{begin-eqn}}
{{eqn | l = \cos x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n!} }
| c =
}}
{{eqn | r = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \frac {x^6} {6!} + \cdots + \paren {-1}^n... | Euler's Cosine Identity/Real Domain/Proof 1 | https://proofwiki.org/wiki/Euler's_Cosine_Identity | https://proofwiki.org/wiki/Euler's_Cosine_Identity/Real_Domain/Proof_1 | [
"Euler's Identities",
"Euler's Cosine Identity",
"Cosine Function"
] | [] | [
"Definition:Cosine/Real Function",
"Definition:Exponential Function/Real/Power Series Expansion",
"Cosine Function is Absolutely Convergent",
"Definition:Even Integer",
"Definition:Odd Integer"
] |
proofwiki-6455 | Euler's Cosine Identity | :$\cos z = \dfrac {e^{i z} + e^{-i z} } 2$ | Recall Euler's Formula:
:$e^{i x} = \cos x + i \sin x$
Then, starting from the {{RHS}}:
{{begin-eqn}}
{{eqn | l = \frac {e^{i x} + e^{-i x} } 2
| r = \frac {\cos x + i \sin x + \map \cos {-x} + i \map \sin {-x} } 2
}}
{{eqn | r = \frac {\cos x + \map \cos {-x} } 2
| c = Sine Function is Odd
}}
{{eqn | r = \... | :$\cos z = \dfrac {e^{i z} + e^{-i z} } 2$ | Recall [[Euler's Formula/Real Domain|Euler's Formula]]:
:$e^{i x} = \cos x + i \sin x$
Then, starting from the {{RHS}}:
{{begin-eqn}}
{{eqn | l = \frac {e^{i x} + e^{-i x} } 2
| r = \frac {\cos x + i \sin x + \map \cos {-x} + i \map \sin {-x} } 2
}}
{{eqn | r = \frac {\cos x + \map \cos {-x} } 2
| c = [[... | Euler's Cosine Identity/Real Domain/Proof 2 | https://proofwiki.org/wiki/Euler's_Cosine_Identity | https://proofwiki.org/wiki/Euler's_Cosine_Identity/Real_Domain/Proof_2 | [
"Euler's Identities",
"Euler's Cosine Identity",
"Cosine Function"
] | [] | [
"Euler's Formula/Real Domain",
"Sine Function is Odd",
"Cosine Function is Even"
] |
proofwiki-6456 | Euler's Cosine Identity | :$\cos z = \dfrac {e^{i z} + e^{-i z} } 2$ | {{begin-eqn}}
{{eqn | n = 1
| l = e^{i x}
| r = \cos x + i \sin x
| c = Euler's Formula
}}
{{eqn | n = 2
| l = e^{-i x}
| r = \cos x - i \sin x
| c = Euler's Formula: Corollary
}}
{{eqn | ll= \leadsto
| l = e^{i x} + e^{-i x}
| r = \paren {\cos x + i \sin x} + \paren {\co... | :$\cos z = \dfrac {e^{i z} + e^{-i z} } 2$ | {{begin-eqn}}
{{eqn | n = 1
| l = e^{i x}
| r = \cos x + i \sin x
| c = [[Euler's Formula/Real Domain|Euler's Formula]]
}}
{{eqn | n = 2
| l = e^{-i x}
| r = \cos x - i \sin x
| c = [[Euler's Formula/Real Domain/Corollary|Euler's Formula: Corollary]]
}}
{{eqn | ll= \leadsto
| l... | Euler's Cosine Identity/Real Domain/Proof 3 | https://proofwiki.org/wiki/Euler's_Cosine_Identity | https://proofwiki.org/wiki/Euler's_Cosine_Identity/Real_Domain/Proof_3 | [
"Euler's Identities",
"Euler's Cosine Identity",
"Cosine Function"
] | [] | [
"Euler's Formula/Real Domain",
"Euler's Formula/Real Domain/Corollary"
] |
proofwiki-6457 | Euler's Cosine Identity | :$\cos z = \dfrac {e^{i z} + e^{-i z} } 2$ | Consider the differential equation:
:$(1): \quad D^2_x \map f x = -\map f x$
subject to the initial conditions:
:$(2): \quad \map f 0 = 1$
:$(3): \quad D_x \map f 0 = 0$
=== Step 1 ===
We will prove that $y = \cos x$ is a particular solution of $(1)$.
{{begin-eqn}}
{{eqn | l = y
| r = \cos x
| c =
}}
{{eqn... | :$\cos z = \dfrac {e^{i z} + e^{-i z} } 2$ | Consider the [[Definition:Second Order Ordinary Differential Equation|differential equation]]:
:$(1): \quad D^2_x \map f x = -\map f x$
subject to the [[Definition:Initial Condition|initial conditions]]:
:$(2): \quad \map f 0 = 1$
:$(3): \quad D_x \map f 0 = 0$
=== Step 1 ===
We will prove that $y = \cos x$ is a [... | Euler's Cosine Identity/Real Domain/Proof 4 | https://proofwiki.org/wiki/Euler's_Cosine_Identity | https://proofwiki.org/wiki/Euler's_Cosine_Identity/Real_Domain/Proof_4 | [
"Euler's Identities",
"Euler's Cosine Identity",
"Cosine Function"
] | [] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Initial Condition",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Derivative/Higher Derivatives/Second Derivative",
"Derivative of Cosine Function",
"Derivative of Constant Multiple",
"Derivative of Sine... |
proofwiki-6458 | Reciprocal of i | :$\dfrac 1 i = -i$ | {{begin-eqn}}
{{eqn | l = i^2
| r = -1
| c = {{Defof|Imaginary Unit}}
}}
{{eqn | ll= \leadsto
| l = \frac {i^2} i
| r = \frac {-1} i
}}
{{eqn | ll= \leadsto
| l = i
| r = \frac {-1} i
}}
{{eqn | ll= \leadsto
| l = -i
| r = \frac 1 i
}}
{{end-eqn}}
{{qed}}
Category:Imagina... | :$\dfrac 1 i = -i$ | {{begin-eqn}}
{{eqn | l = i^2
| r = -1
| c = {{Defof|Imaginary Unit}}
}}
{{eqn | ll= \leadsto
| l = \frac {i^2} i
| r = \frac {-1} i
}}
{{eqn | ll= \leadsto
| l = i
| r = \frac {-1} i
}}
{{eqn | ll= \leadsto
| l = -i
| r = \frac 1 i
}}
{{end-eqn}}
{{qed}}
[[Category:Imag... | Reciprocal of i | https://proofwiki.org/wiki/Reciprocal_of_i | https://proofwiki.org/wiki/Reciprocal_of_i | [
"Imaginary Unit",
"Examples of Reciprocals"
] | [] | [
"Category:Imaginary Unit",
"Category:Examples of Reciprocals"
] |
proofwiki-6459 | Cosine of Difference | :$\map \cos {a - b} = \cos a \cos b + \sin a \sin b$ | {{begin-eqn}}
{{eqn | l = \map \cos {a - b}
| r = \cos a \map \cos {-b} - \sin a \map \sin {-b}
| c = Cosine of Sum
}}
{{eqn | r = \cos a \cos b - \sin a \map \sin {-b}
| c = Cosine Function is Even
}}
{{eqn | r = \cos a \cos b + \sin a \sin b
| c = Sine Function is Odd
}}
{{end-eqn}}
{{qed}} | :$\map \cos {a - b} = \cos a \cos b + \sin a \sin b$ | {{begin-eqn}}
{{eqn | l = \map \cos {a - b}
| r = \cos a \map \cos {-b} - \sin a \map \sin {-b}
| c = [[Cosine of Sum]]
}}
{{eqn | r = \cos a \cos b - \sin a \map \sin {-b}
| c = [[Cosine Function is Even]]
}}
{{eqn | r = \cos a \cos b + \sin a \sin b
| c = [[Sine Function is Odd]]
}}
{{end-eqn}... | Cosine of Difference/Proof 1 | https://proofwiki.org/wiki/Cosine_of_Difference | https://proofwiki.org/wiki/Cosine_of_Difference/Proof_1 | [
"Cosine of Difference",
"Cosine Function",
"Trigonometric Subtraction Formulas"
] | [] | [
"Cosine of Sum",
"Cosine Function is Even",
"Sine Function is Odd"
] |
proofwiki-6460 | Rational Numbers are not Connected | The set of rational numbers $\Q$ is not a connected topological space. | Let $\alpha \in \R$ be an irrational number.
By definition, $\alpha \notin \Q$.
Consider the sets:
:$S := \Q \cap \openint \gets \alpha$
:$T := \Q \cap \openint \alpha \to$
Let $x \in S$.
Let $\map {B_\epsilon} x$ be the open $\epsilon$-ball of $x$ in $\Q$.
Then:
:$\forall x \in S: \exists \epsilon \in \R_{>0}: \map {B... | The [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]] $\Q$ is not a [[Definition:Connected Topological Space|connected topological space]]. | Let $\alpha \in \R$ be an [[Definition:Irrational Number|irrational number]].
By definition, $\alpha \notin \Q$.
Consider the [[Definition:Set|sets]]:
:$S := \Q \cap \openint \gets \alpha$
:$T := \Q \cap \openint \alpha \to$
Let $x \in S$.
Let $\map {B_\epsilon} x$ be the [[Definition:Open Ball|open $\epsilon$-ball... | Rational Numbers are not Connected | https://proofwiki.org/wiki/Rational_Numbers_are_not_Connected | https://proofwiki.org/wiki/Rational_Numbers_are_not_Connected | [
"Rational Numbers",
"Examples of Connected Topological Spaces"
] | [
"Definition:Set",
"Definition:Rational Number",
"Definition:Connected Topological Space"
] | [
"Definition:Irrational Number",
"Definition:Set",
"Definition:Open Ball",
"Definition:Open Set/Metric Space",
"Definition:Separation (Topology)",
"Definition:Connected Topological Space"
] |
proofwiki-6461 | Closure of Connected Set is Connected | Let $T$ be a topological space.
Let $H$ be a connected set of $T$.
Let $H^-$ denote the closure of $H$ in $T$.
Then $H^-$ is connected in $T$. | By Set is Subset of Itself, the result follows by setting $K = H^-$ in Set between Connected Set and Closure is Connected.
{{qed}} | Let $T$ be a [[Definition:Topological Space|topological space]].
Let $H$ be a [[Definition:Connected Set (Topology)|connected set]] of $T$.
Let $H^-$ denote the [[Definition:Closure (Topology)|closure]] of $H$ in $T$.
Then $H^-$ is [[Definition:Connected Set (Topology)|connected]] in $T$. | By [[Set is Subset of Itself]], the result follows by setting $K = H^-$ in [[Set between Connected Set and Closure is Connected]].
{{qed}} | Closure of Connected Set is Connected | https://proofwiki.org/wiki/Closure_of_Connected_Set_is_Connected | https://proofwiki.org/wiki/Closure_of_Connected_Set_is_Connected | [
"Set Closures",
"Connected Sets (Topology)"
] | [
"Definition:Topological Space",
"Definition:Connected Set (Topology)",
"Definition:Closure (Topology)",
"Definition:Connected Set (Topology)"
] | [
"Set is Subset of Itself",
"Set between Connected Set and Closure is Connected"
] |
proofwiki-6462 | Closed Topologist's Sine Curve is Connected | Let $T$ be the closed topologist's sine curve.
Then $T$ is connected. | {{Recall|Closed Topologist's Sine Curve}}
{{:Definition:Closed Topologist's Sine Curve}}
Because the open interval $\openint 0 \infty$ is connected, then so is $G$ by Continuous Image of Connected Space is Connected.
It is enough, from Set between Connected Set and Closure is Connected, to show that $J \subseteq \map \... | Let $T$ be the [[Definition:Closed Topologist's Sine Curve|closed topologist's sine curve]].
Then $T$ is [[Definition:Connected Topological Space|connected]]. | {{Recall|Closed Topologist's Sine Curve}}
{{:Definition:Closed Topologist's Sine Curve}}
Because the [[Definition:Open Real Interval|open interval]] $\openint 0 \infty$ is [[Definition:Connected Topological Space|connected]], then so is $G$ by [[Continuous Image of Connected Space is Connected]].
It is enough, from ... | Closed Topologist's Sine Curve is Connected | https://proofwiki.org/wiki/Closed_Topologist's_Sine_Curve_is_Connected | https://proofwiki.org/wiki/Closed_Topologist's_Sine_Curve_is_Connected | [
"Closed Topologist's Sine Curve",
"Examples of Connected Topological Spaces"
] | [
"Definition:Closed Topologist's Sine Curve",
"Definition:Connected Topological Space"
] | [
"Definition:Real Interval/Open",
"Definition:Connected Topological Space",
"Continuous Image of Connected Space is Connected",
"Set between Connected Set and Closure is Connected",
"Definition:Neighborhood (Real Analysis)/Epsilon",
"Sine of Half-Integer Multiple of Pi",
"Intermediate Value Theorem",
"... |
proofwiki-6463 | Separated Sets are Clopen in Union | Let $T = \left({S, \tau}\right)$ be a topological space.
Let $A$ and $B$ be separated sets in $T$.
Let $H = A \cup B$ be given the subspace topology.
Then $A$ and $B$ are each both open and closed in $H$. | By hypothesis, $A$ and $B$ are separated:
:$A \cap B^- = A^- \cap B = \O$
Then:
{{begin-eqn}}
{{eqn | l = H \cap B^-
| r = \paren {A \cup B} \cap B^-
}}
{{eqn | r = \paren {A \cap B^-} \cup \paren {B \cap B^-}
| c = Intersection Absorbs Union
}}
{{eqn | r = \O \cup B
| c = Set is Subset of its Topolog... | Let $T = \left({S, \tau}\right)$ be a [[Definition:Topological Space|topological space]].
Let $A$ and $B$ be [[Definition:Separated Sets|separated sets]] in $T$.
Let $H = A \cup B$ be given the [[Definition:Subspace Topology|subspace topology]].
Then $A$ and $B$ are each [[Definition:Clopen Set|both open and closed]... | [[Definition:By Hypothesis|By hypothesis]], $A$ and $B$ are [[Definition:Separated Sets|separated]]:
:$A \cap B^- = A^- \cap B = \O$
Then:
{{begin-eqn}}
{{eqn | l = H \cap B^-
| r = \paren {A \cup B} \cap B^-
}}
{{eqn | r = \paren {A \cap B^-} \cup \paren {B \cap B^-}
| c = [[Intersection Absorbs Union]]
}... | Separated Sets are Clopen in Union | https://proofwiki.org/wiki/Separated_Sets_are_Clopen_in_Union | https://proofwiki.org/wiki/Separated_Sets_are_Clopen_in_Union | [
"Separated Sets"
] | [
"Definition:Topological Space",
"Definition:Separated Sets",
"Definition:Topological Subspace",
"Definition:Clopen Set"
] | [
"Definition:By Hypothesis",
"Definition:Separated Sets",
"Absorption Laws (Set Theory)/Intersection Absorbs Union",
"Set is Subset of its Topological Closure",
"Intersection with Subset is Subset",
"Union with Empty Set",
"Definition:Set Intersection",
"Definition:Closed Set/Topology",
"Definition:T... |
proofwiki-6464 | Compatibility of Atlases is Equivalence Relation | Let $M$ be a topological space.
Let $d$ and $k$ be natural numbers.
Let $\AA$ denote the set of all $d$-dimensional atlases of class $\CC^k$ on $M$.
Define a relation $\sim$ on $\AA$ by putting, for any two $\CC^k$-atlases $\FF$ and $\GG$:
:$\FF \sim \GG$ {{iff}} $\FF$ and $\GG$ are $C^k$-compatible.
Then $\sim$ is an ... | It is to be shown that $\sim$ is reflexive, symmetric and transitive. | Let $M$ be a [[Definition:Topological Space|topological space]].
Let $d$ and $k$ be [[Definition:Natural Number|natural numbers]].
Let $\AA$ denote the [[Definition:Set|set]] of all $d$-[[Definition:Dimension of Atlas|dimensional]] [[Definition:Atlas|atlases]] of [[Definition:Class of Atlas|class]] $\CC^k$ on $M$.
D... | It is to be shown that $\sim$ is [[Definition:Reflexive Relation|reflexive]], [[Definition:Symmetric Relation|symmetric]] and [[Definition:Transitive Relation|transitive]]. | Compatibility of Atlases is Equivalence Relation | https://proofwiki.org/wiki/Compatibility_of_Atlases_is_Equivalence_Relation | https://proofwiki.org/wiki/Compatibility_of_Atlases_is_Equivalence_Relation | [
"Compatible Atlases",
"Examples of Equivalence Relations"
] | [
"Definition:Topological Space",
"Definition:Natural Numbers",
"Definition:Set",
"Definition:Atlas",
"Definition:Atlas",
"Definition:Atlas",
"Definition:Relation",
"Definition:Atlas",
"Definition:Compatible Atlases",
"Definition:Equivalence Relation"
] | [
"Definition:Reflexive Relation",
"Definition:Symmetric Relation",
"Definition:Transitive Relation"
] |
proofwiki-6465 | Addition of Real and Imaginary Parts | Let $z_0, z_1 \in \C$ be two complex numbers.
Then:
:$\map \Re {z_0 + z_1} = \map \Re {z_0} + \map \Re {z_1}$
and:
:$\map \Im {z_0 + z_1} = \map \Im {z_0} + \map \Im {z_1}$
Here, $\map \Re {z_0}$ denotes the real part of $z_0$, and $\map \Im {z_0}$ denotes the imaginary part of $z_0$. | We have:
{{begin-eqn}}
{{eqn | l = z_0 + z_1
| r = \paren {\map \Re {z_0} + i \, \map \Im {z_0} } + \paren {\map \Re {z_1} + i \, \map \Im {z_1} }
| c = {{Defof|Complex Number}}
}}
{{eqn | r = \paren {\map \Re {z_0} + \map \Re {z_1} } + i \paren {\map \Im {z_0} + \map \Im {z_1} }
| c = {{Defof|Complex... | Let $z_0, z_1 \in \C$ be two [[Definition:Complex Number|complex numbers]].
Then:
:$\map \Re {z_0 + z_1} = \map \Re {z_0} + \map \Re {z_1}$
and:
:$\map \Im {z_0 + z_1} = \map \Im {z_0} + \map \Im {z_1}$
Here, $\map \Re {z_0}$ denotes the [[Definition:Real Part|real part]] of $z_0$, and $\map \Im {z_0}$ denotes t... | We have:
{{begin-eqn}}
{{eqn | l = z_0 + z_1
| r = \paren {\map \Re {z_0} + i \, \map \Im {z_0} } + \paren {\map \Re {z_1} + i \, \map \Im {z_1} }
| c = {{Defof|Complex Number}}
}}
{{eqn | r = \paren {\map \Re {z_0} + \map \Re {z_1} } + i \paren {\map \Im {z_0} + \map \Im {z_1} }
| c = {{Defof|Comple... | Addition of Real and Imaginary Parts | https://proofwiki.org/wiki/Addition_of_Real_and_Imaginary_Parts | https://proofwiki.org/wiki/Addition_of_Real_and_Imaginary_Parts | [
"Complex Addition"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Real Part",
"Definition:Complex Number/Imaginary Part"
] | [
"Category:Complex Addition"
] |
proofwiki-6466 | Multiplication of Real and Imaginary Parts | Let $w, z \in \C$ be complex numbers.
$(1)$ If $w$ is wholly real, then:
:$\map \Re {w z} = w \, \map \Re z$
and:
:$\map \Im {w z} = w \, \map \Im z$
$(2)$ If $w$ is wholly imaginary, then:
:$\map \Re {w z} = -\map \Im w \, \map \Im z$
and:
:$\map \Im {w z} = \map \Im w \, \map \Re z$
Here, $\map \Re z$ denotes the rea... | Assume that $w$ is wholly real.
Then:
{{begin-eqn}}
{{eqn | l = w z
| r = \map \Re w \, \map \Re z - \map \Im w \, \map \Im z + i \paren {\map \Re w \, \map \Im z + \map \Im w \, \map \Re z}
| c = {{Defof|Complex Multiplication}}
}}
{{eqn | r = w \, \map \Re z + i w \, \map \Im z
| c = as $\map \Re w... | Let $w, z \in \C$ be [[Definition:Complex Number|complex numbers]].
$(1)$ If $w$ is [[Definition:Wholly Real|wholly real]], then:
:$\map \Re {w z} = w \, \map \Re z$
and:
:$\map \Im {w z} = w \, \map \Im z$
$(2)$ If $w$ is [[Definition:Wholly Imaginary|wholly imaginary]], then:
:$\map \Re {w z} = -\map \Im w \,... | Assume that $w$ is [[Definition:Wholly Real|wholly real]].
Then:
{{begin-eqn}}
{{eqn | l = w z
| r = \map \Re w \, \map \Re z - \map \Im w \, \map \Im z + i \paren {\map \Re w \, \map \Im z + \map \Im w \, \map \Re z}
| c = {{Defof|Complex Multiplication}}
}}
{{eqn | r = w \, \map \Re z + i w \, \map \Im ... | Multiplication of Real and Imaginary Parts | https://proofwiki.org/wiki/Multiplication_of_Real_and_Imaginary_Parts | https://proofwiki.org/wiki/Multiplication_of_Real_and_Imaginary_Parts | [
"Complex Multiplication"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Wholly Real",
"Definition:Complex Number/Wholly Imaginary",
"Definition:Complex Number/Real Part",
"Definition:Complex Number/Imaginary Part"
] | [
"Definition:Complex Number/Wholly Real",
"Definition:Complex Number/Wholly Imaginary",
"Category:Complex Multiplication"
] |
proofwiki-6467 | Rational Numbers are Totally Disconnected | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Then $\struct {\Q, \tau_d}$ is a totally disconnected space. | Let $x, y \in \Q$ such that $x \ne y$.
From Between two Rational Numbers exists Irrational Number, there exists $\alpha \in \R \setminus \Q$ such that $x < \alpha < y$.
From Rational Numbers are not Connected, it follows that $x$ and $y$ belong to different components of $\Q$.
As $x$ and $y$ are arbitrary, it follows t... | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\Q, \tau_d}$ is a [[Definition:Totally Disconnected Space|totally disconnected space]]. | Let $x, y \in \Q$ such that $x \ne y$.
From [[Between two Rational Numbers exists Irrational Number]], there exists $\alpha \in \R \setminus \Q$ such that $x < \alpha < y$.
From [[Rational Numbers are not Connected]], it follows that $x$ and $y$ belong to different [[Definition:Component (Topology)|components]] of $\... | Rational Numbers are Totally Disconnected/Proof 1 | https://proofwiki.org/wiki/Rational_Numbers_are_Totally_Disconnected | https://proofwiki.org/wiki/Rational_Numbers_are_Totally_Disconnected/Proof_1 | [
"Rational Numbers are Totally Disconnected",
"Rational Numbers",
"Examples of Totally Disconnected Spaces"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Totally Disconnected Space"
] | [
"Between two Rational Numbers exists Irrational Number",
"Rational Numbers are not Connected",
"Definition:Component (Topology)",
"Definition:Rational Number",
"Definition:Component (Topology)",
"Definition:Rational Number",
"Definition:Component (Topology)",
"Definition:Singleton",
"Definition:Tota... |
proofwiki-6468 | Rational Numbers are Totally Disconnected | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Then $\struct {\Q, \tau_d}$ is a totally disconnected space. | Follows from:
: Rational Number Space is Totally Separated
: Totally Separated Space is Totally Disconnected
{{qed}} | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\Q, \tau_d}$ is a [[Definition:Totally Disconnected Space|totally disconnected space]]. | Follows from:
: [[Rational Number Space is Totally Separated]]
: [[Totally Separated Space is Totally Disconnected]]
{{qed}} | Rational Numbers are Totally Disconnected/Proof 2 | https://proofwiki.org/wiki/Rational_Numbers_are_Totally_Disconnected | https://proofwiki.org/wiki/Rational_Numbers_are_Totally_Disconnected/Proof_2 | [
"Rational Numbers are Totally Disconnected",
"Rational Numbers",
"Examples of Totally Disconnected Spaces"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Totally Disconnected Space"
] | [
"Rational Number Space is Totally Separated",
"Totally Separated Space is Totally Disconnected"
] |
proofwiki-6469 | Discrete Space is Totally Disconnected | Let $T = \struct {S, \tau}$ be a topological space where $\tau$ is the discrete topology on $S$.
Then $T$ is totally disconnected. | Follows from:
* Discrete Space is Extremally Disconnected Hausdorff
* Extremally Disconnected Hausdorff Space is Totally Separated
* Totally Separated Space is Totally Disconnected
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] where $\tau$ is the [[Definition:Discrete Topology|discrete topology]] on $S$.
Then $T$ is [[Definition:Totally Disconnected Space|totally disconnected]]. | Follows from:
* [[Discrete Space is Extremally Disconnected Hausdorff]]
* [[Extremally Disconnected Hausdorff Space is Totally Separated]]
* [[Totally Separated Space is Totally Disconnected]]
{{qed}} | Discrete Space is Totally Disconnected | https://proofwiki.org/wiki/Discrete_Space_is_Totally_Disconnected | https://proofwiki.org/wiki/Discrete_Space_is_Totally_Disconnected | [
"Discrete Topologies",
"Examples of Totally Disconnected Spaces"
] | [
"Definition:Topological Space",
"Definition:Discrete Topology",
"Definition:Totally Disconnected Space"
] | [
"Discrete Space is Extremally Disconnected Hausdorff",
"Extremally Disconnected Hausdorff Space is Totally Separated",
"Totally Separated Space is Totally Disconnected"
] |
proofwiki-6470 | Rational Numbers are not Discrete Space | The set of rational numbers $\Q$ does not form a discrete space. | Follows from:
:Rational Number Space is not Extremally Disconnected
:Discrete Space is Extremally Disconnected Hausdorff.
{{qed}} | The [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]] $\Q$ does not form a [[Definition:Discrete Space|discrete space]]. | Follows from:
:[[Rational Number Space is not Extremally Disconnected]]
:[[Discrete Space is Extremally Disconnected Hausdorff]].
{{qed}} | Rational Numbers are not Discrete Space | https://proofwiki.org/wiki/Rational_Numbers_are_not_Discrete_Space | https://proofwiki.org/wiki/Rational_Numbers_are_not_Discrete_Space | [
"Rational Numbers",
"Discrete Topologies"
] | [
"Definition:Set",
"Definition:Rational Number",
"Definition:Discrete Topology"
] | [
"Rational Number Space is not Extremally Disconnected",
"Discrete Space is Extremally Disconnected Hausdorff"
] |
proofwiki-6471 | Rational Number Space is not Extremally Disconnected | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Then $\struct {\Q, \tau_d}$ does not form an extremally disconnected space. | {{Recall|Extremally Disconnected Space|extremally disconnected space}}
{{Definition:Extremally Disconnected Space/Definition 1}}
Hence the existence will be demonstrated of an open set in $\struct {\Q, \tau_d}$ whose closure is not open.
We have that $\openint 0 1$ is open in $\struct {\R, \tau_d}$.
Thus $\openint 0 1 ... | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then $\struct {\Q, \tau_d}$ does not form an [[Definition:Extremally Disconnected Space|extremally disconnected space]]. | {{Recall|Extremally Disconnected Space|extremally disconnected space}}
{{Definition:Extremally Disconnected Space/Definition 1}}
Hence the existence will be demonstrated of an [[Definition:Open Set (Topology)|open set]] in $\struct {\Q, \tau_d}$ whose [[Definition:Closure (Topology)|closure]] is not [[Definition:Open ... | Rational Number Space is not Extremally Disconnected | https://proofwiki.org/wiki/Rational_Number_Space_is_not_Extremally_Disconnected | https://proofwiki.org/wiki/Rational_Number_Space_is_not_Extremally_Disconnected | [
"Rational Number Space",
"Extremally Disconnected Spaces"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Extremally Disconnected Space"
] | [
"Definition:Open Set/Topology",
"Definition:Closure (Topology)",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Limit Point/Topology/Set",
"Between two Real Numbers exists Rational Number",
"Definition:Limit Point/Topology/Set",
"Definitio... |
proofwiki-6472 | Points in Product Spaces are Near Open Sets | Let $\family {X_i}_{i \mathop \in I}$ be an indexed family of topological spaces, where $I$ is an arbitrary index set.
Let $X = \ds \prod_{i \mathop \in I} X_i$ be the product space of $\family {X_i}_{i \mathop \in I}$.
Let $U$ be nonempty open subset of $X$.
Let $x$ be a point in $X$.
For each point $y$ in $X$, let $\... | Let $q$ be any point in $U$.
The topology on the product space $X$ is the product topology which has the natural basis as a (synthetic) basis.
Then:
:$\exists Q \in \BB : q \in Q \subseteq U$
where $\BB$ is the natural basis for product topology on $X$.
From Natural Basis of Product Topology:
:$\ds Q = \prod_{i \mathop... | Let $\family {X_i}_{i \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Topological Space|topological spaces]], where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $X = \ds \prod_{i \mathop \in I} X_i$ be the [[Definition:Product Space of Topological Spaces|product spa... | Let $q$ be any [[Definition:Element|point]] in $U$.
The [[Definition:Topology|topology]] on the [[Definition:Product Space of Topological Spaces|product space]] $X$ is the [[Definition:Product Topology|product topology]] which has the [[Definition:Natural Basis of Product Topology|natural basis]] as a [[Definition:Syn... | Points in Product Spaces are Near Open Sets | https://proofwiki.org/wiki/Points_in_Product_Spaces_are_Near_Open_Sets | https://proofwiki.org/wiki/Points_in_Product_Spaces_are_Near_Open_Sets | [
"Product Topology"
] | [
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:Non-Empty Set",
"Definition:Open Set",
"Definition:Element",
"Definition:Element",
"Definition:Element",
"Definition:Finite Set"
] | [
"Definition:Element",
"Definition:Topology",
"Definition:Product Space (Topology)",
"Definition:Product Topology",
"Definition:Product Topology/Natural Basis",
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Product Topology/Natural Basis",
"Definition:Product Topology",
"Natural Basis of... |
proofwiki-6473 | Real and Imaginary Part Projections are Continuous | Define the real-valued functions $x, y: \C \to \R$ by:
:$\forall z \in \C: \map x z = \map \Re z$
:$\forall z \in \C: \map y z = \map \Im z$
Equip $\R$ with the usual Euclidean metric.
Equip $\C$ with the usual Euclidean metric.
Then both $x$ and $y$ are continuous functions. | Let $z \in \C$, and let $\epsilon \in \R_{>0}$.
Put $\delta = \epsilon$.
For all $w \in \C$ with $\cmod {w - z} < \delta$:
{{begin-eqn}}
{{eqn | l = \cmod {\map \Re w - \map \Re z}
| r = \cmod {\map \Re w + i \map \Im w - \map \Re z - i \map \Im z + i \map \Im z - i \map \Im w}
}}
{{eqn | o = \le
| r = \cmo... | Define the [[Definition:Real-Valued Function|real-valued functions]] $x, y: \C \to \R$ by:
:$\forall z \in \C: \map x z = \map \Re z$
:$\forall z \in \C: \map y z = \map \Im z$
Equip $\R$ with the usual [[Definition:Euclidean Metric on Real Number Line|Euclidean metric]].
Equip $\C$ with the usual [[Definition:Euc... | Let $z \in \C$, and let $\epsilon \in \R_{>0}$.
Put $\delta = \epsilon$.
For all $w \in \C$ with $\cmod {w - z} < \delta$:
{{begin-eqn}}
{{eqn | l = \cmod {\map \Re w - \map \Re z}
| r = \cmod {\map \Re w + i \map \Im w - \map \Re z - i \map \Im z + i \map \Im z - i \map \Im w}
}}
{{eqn | o = \le
| r = \... | Real and Imaginary Part Projections are Continuous | https://proofwiki.org/wiki/Real_and_Imaginary_Part_Projections_are_Continuous | https://proofwiki.org/wiki/Real_and_Imaginary_Part_Projections_are_Continuous | [
"Complex Numbers",
"Continuous Functions"
] | [
"Definition:Real-Valued Function",
"Definition:Euclidean Metric/Real Number Line",
"Definition:Euclidean Metric/Complex Plane",
"Definition:Continuous Mapping (Metric Space)/Space"
] | [
"Triangle Inequality/Complex Numbers",
"Complex Modulus is Non-Negative",
"Complex Modulus of Product of Complex Numbers",
"Triangle Inequality/Complex Numbers",
"Complex Modulus is Non-Negative",
"Definition:Continuous Mapping (Metric Space)/Space"
] |
proofwiki-6474 | Continuous Complex Function is Complex Riemann Integrable | Let $\closedint a b$ be a closed real interval.
Let $f: \closedint a b \to \C$ be a continuous complex function.
Then $f$ is complex Riemann integrable over $\closedint a b$. | Define the real function $x: \closedint a b \to \R$ by:
:$\forall t \in \closedint a b : \map x t = \map \Re {\map f t}$
Define the real function $y: \closedint a b \to \R$ by:
:$\forall t \in \closedint a b : \map y t = \map \Im {\map f t}$
where:
:$\map \Re {\map f t}$ denotes the real part of the complex number $\ma... | Let $\closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $f: \closedint a b \to \C$ be a [[Definition:Continuous Complex Function|continuous]] [[Definition:Complex Function|complex function]].
Then $f$ is [[Definition:Complex Riemann Integral|complex Riemann integrable]] over $\closedi... | Define the [[Definition:Real Function|real function]] $x: \closedint a b \to \R$ by:
:$\forall t \in \closedint a b : \map x t = \map \Re {\map f t}$
Define the real function $y: \closedint a b \to \R$ by:
:$\forall t \in \closedint a b : \map y t = \map \Im {\map f t}$
where:
:$\map \Re {\map f t}$ denotes the [[D... | Continuous Complex Function is Complex Riemann Integrable | https://proofwiki.org/wiki/Continuous_Complex_Function_is_Complex_Riemann_Integrable | https://proofwiki.org/wiki/Continuous_Complex_Function_is_Complex_Riemann_Integrable | [
"Complex Analysis"
] | [
"Definition:Real Interval/Closed",
"Definition:Continuous Complex Function",
"Definition:Complex Function",
"Definition:Integrable Function/Complex"
] | [
"Definition:Real Function",
"Definition:Complex Number/Real Part",
"Definition:Complex Number",
"Definition:Complex Number/Imaginary Part",
"Real and Imaginary Part Projections are Continuous",
"Definition:Continuous Mapping (Metric Space)",
"Composite of Continuous Mappings is Continuous",
"Continuou... |
proofwiki-6475 | Compact Subspace of Metric Space is Sequentially Compact in Itself | Let $M = \struct {A, d}$ be a metric space.
Let $C \subseteq M$ be a subspace of $M$ such that $C$ is compact.
Then $C$ is sequentially compact in itself. | Let $C \subseteq M$ be compact.
Let $\sequence {x_n}$ be a sequence in $C$.
Let $S$ be the range of $\sequence {x_n}$.
Thus $S \subseteq C$ and $S$ may be either finite or infinite. | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $C \subseteq M$ be a [[Definition:Metric Subspace|subspace]] of $M$ such that $C$ is [[Definition:Compact Metric Space|compact]].
Then $C$ is [[Definition:Sequentially Compact In Itself|sequentially compact in itself]]. | Let $C \subseteq M$ be [[Definition:Compact Metric Space|compact]].
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence in $C$]].
Let $S$ be the [[Definition:Range of Sequence|range of $\sequence {x_n}$]].
Thus $S \subseteq C$ and $S$ may be either [[Definition:Finite Set|finite]] or [[Definition:Infinite Set|... | Compact Subspace of Metric Space is Sequentially Compact in Itself | https://proofwiki.org/wiki/Compact_Subspace_of_Metric_Space_is_Sequentially_Compact_in_Itself | https://proofwiki.org/wiki/Compact_Subspace_of_Metric_Space_is_Sequentially_Compact_in_Itself | [
"Compact Metric Spaces",
"Sequentially Compact Spaces",
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Metric Subspace",
"Definition:Compact Space/Metric Space",
"Definition:Sequentially Compact Space/In Itself"
] | [
"Definition:Compact Space/Metric Space",
"Definition:Sequence",
"Definition:Range of Sequence",
"Definition:Finite Set",
"Definition:Infinite Set",
"Definition:Finite Set",
"Definition:Infinite Set",
"Definition:Compact Space/Metric Space",
"Definition:Finite Set",
"Definition:Finite Set"
] |
proofwiki-6476 | Products of Products are Homeomorphic to Collapsed Products | Let $I$ be an index set, and for each $i \in I$ let $J_i$ be an index set.
Let the sets $J_i$ be pairwise disjoint.
Let $\ds J = \bigcup_{i \mathop \in I} J_i$
For each $j \in J$, let $X_j$ be a topological space.
Then $\ds \prod_{j \mathop \in J} X_j$ is homeomorphic to $\ds \prod_{i \mathop \in I} \prod_{j \mathop \i... | {{proof wanted}}
Category:Product Spaces
0s9l5qbbx5pb7ixmlpcjy0sjn45jpqn | Let $I$ be an index set, and for each $i \in I$ let $J_i$ be an index set.
Let the sets $J_i$ be pairwise disjoint.
Let $\ds J = \bigcup_{i \mathop \in I} J_i$
For each $j \in J$, let $X_j$ be a topological space.
Then $\ds \prod_{j \mathop \in J} X_j$ is homeomorphic to $\ds \prod_{i \mathop \in I} \prod_{j \matho... | {{proof wanted}}
[[Category:Product Spaces]]
0s9l5qbbx5pb7ixmlpcjy0sjn45jpqn | Products of Products are Homeomorphic to Collapsed Products | https://proofwiki.org/wiki/Products_of_Products_are_Homeomorphic_to_Collapsed_Products | https://proofwiki.org/wiki/Products_of_Products_are_Homeomorphic_to_Collapsed_Products | [
"Product Spaces"
] | [] | [
"Category:Product Spaces"
] |
proofwiki-6477 | Sum of Complex Integrals on Adjacent Intervals | Let $\closedint a b$ be a closed real interval.
Let $f: \closedint a b \to \C$ be a continuous complex function.
Let $c \in \closedint a b$.
Then:
:$\ds \int_a^c \map f t \rd t + \int_c^b \map f t \rd t = \int_a^b \map f t \rd t$ | From Continuous Complex Function is Complex Riemann Integrable, it follows that all three complex Riemann integrals are well defined.
From Real and Imaginary Part Projections are Continuous, it follows that $\Re: \C \to \R$ and $\Im: \C \to \R$ are continuous functions.
{{explain|Revisit the above link -- see if there ... | Let $\closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $f: \closedint a b \to \C$ be a [[Definition:Continuous Complex Function|continuous complex function]].
Let $c \in \closedint a b$.
Then:
:$\ds \int_a^c \map f t \rd t + \int_c^b \map f t \rd t = \int_a^b \map f t \rd t$ | From [[Continuous Complex Function is Complex Riemann Integrable]], it follows that all three [[Definition:Complex Riemann Integral|complex Riemann integrals]] are well defined.
From [[Real and Imaginary Part Projections are Continuous]], it follows that $\Re: \C \to \R$ and $\Im: \C \to \R$ are [[Definition:Continuou... | Sum of Complex Integrals on Adjacent Intervals | https://proofwiki.org/wiki/Sum_of_Complex_Integrals_on_Adjacent_Intervals | https://proofwiki.org/wiki/Sum_of_Complex_Integrals_on_Adjacent_Intervals | [
"Complex Analysis"
] | [
"Definition:Real Interval/Closed",
"Definition:Continuous Complex Function"
] | [
"Continuous Complex Function is Complex Riemann Integrable",
"Definition:Integrable Function/Complex",
"Real and Imaginary Part Projections are Continuous",
"Definition:Continuous Mapping (Metric Space)",
"Composite of Continuous Mappings is Continuous",
"Definition:Continuous Real Function",
"Sum of In... |
proofwiki-6478 | Equivalence of Definitions of Metric Space Continuity at Point | 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|continuity at a point}} | === $\epsilon$-$\delta$ Definition iff Definition by Limits ===
This is proved in Metric Space Continuity by Epsilon-Delta.
{{qed|lemma}} | 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|continuity at a point}} | === $\epsilon$-$\delta$ Definition iff Definition by Limits ===
This is proved in [[Metric Space Continuity by Epsilon-Delta]].
{{qed|lemma}} | Equivalence of Definitions of Metric Space Continuity at Point | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Metric_Space_Continuity_at_Point | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Metric_Space_Continuity_at_Point | [
"Continuous Mappings on Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Mapping"
] | [
"Metric Space Continuity by Epsilon-Delta"
] |
proofwiki-6479 | Equivalence of Definitions of Limit of Mapping between Metric Spaces | {{TFAE|def = Limit of Mapping between Metric Spaces}}
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $c$ be a limit point of $M_1$.
Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$ defined everywhere on $A_1$ ''except possibly'' at $c$.
Let $L \in M_2$. | === $\epsilon$-$\delta$ Condition implies $\epsilon$-Ball Condition ===
Suppose that $f$ satisfies the $\epsilon$-$\delta$ condition:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: 0 < \map {d_1} {x, c} < \delta \implies \map {d_2} {\map f x, L} < \epsilon$
Let $y \in f \sqbrk {\map {B_\delta} {c; d_1} \se... | {{TFAE|def = Limit of Mapping between Metric Spaces}}
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be [[Definition:Metric Space|metric spaces]].
Let $c$ be a [[Definition:Limit Point (Metric Space)|limit point]] of $M_1$.
Let $f: A_1 \to A_2$ be a [[Definition:Mapping|mapping]] from $A_1$ to $A_2$ ... | === $\epsilon$-$\delta$ Condition implies $\epsilon$-Ball Condition ===
Suppose that $f$ satisfies the [[Definition:Limit of Mapping between Metric Spaces/Epsilon-Delta Condition|$\epsilon$-$\delta$ condition]]:
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: 0 < \map {d_1} {x, c} < \delta \implies \map {d... | Equivalence of Definitions of Limit of Mapping between Metric Spaces | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Limit_of_Mapping_between_Metric_Spaces | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Limit_of_Mapping_between_Metric_Spaces | [
"Limits of Mappings between Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Limit Point/Metric Space",
"Definition:Mapping"
] | [
"Definition:Limit of Mapping between Metric Spaces/Epsilon-Delta Condition",
"Definition:Open Ball",
"Definition:By Hypothesis",
"Definition:Subset",
"Definition:Limit of Mapping between Metric Spaces/Epsilon-Ball Condition",
"Definition:Limit of Mapping between Metric Spaces/Epsilon-Ball Condition",
"D... |
proofwiki-6480 | Ordering Principle | Let $S$ be a set.
Then there exists a total ordering on $S$. | From Zermelo's Well-Ordering Theorem, $S$ has a well-ordering.
The result follows from Well-Ordering is Total Ordering.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Then there exists a [[Definition:Total Ordering|total ordering]] on $S$. | From [[Zermelo's Well-Ordering Theorem]], $S$ has a [[Definition:Well-Ordering|well-ordering]].
The result follows from [[Well-Ordering is Total Ordering]].
{{qed}} | Ordering Principle/Proof 1 | https://proofwiki.org/wiki/Ordering_Principle | https://proofwiki.org/wiki/Ordering_Principle/Proof_1 | [
"Set Theory",
"Total Orderings",
"Order Theory",
"Named Theorems",
"Ordering Principle"
] | [
"Definition:Set",
"Definition:Total Ordering"
] | [
"Zermelo's Well-Ordering Theorem",
"Definition:Well-Ordering",
"Equivalence of Definitions of Well-Ordering/Definition 1 implies Definition 2"
] |
proofwiki-6481 | Ordering Principle | Let $S$ be a set.
Then there exists a total ordering on $S$. | This theorem follows trivially from the Order-Extension Principle.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Then there exists a [[Definition:Total Ordering|total ordering]] on $S$. | This theorem follows trivially from the [[Order-Extension Principle]].
{{qed}} | Ordering Principle/Proof 2 | https://proofwiki.org/wiki/Ordering_Principle | https://proofwiki.org/wiki/Ordering_Principle/Proof_2 | [
"Set Theory",
"Total Orderings",
"Order Theory",
"Named Theorems",
"Ordering Principle"
] | [
"Definition:Set",
"Definition:Total Ordering"
] | [
"Order-Extension Principle"
] |
proofwiki-6482 | Ordering Principle | Let $S$ be a set.
Then there exists a total ordering on $S$. | Let $S$ be a non-empty subset of the set of natural numbers $\N$.
We take as axiomatic that $\N$ is itself a subset of the set of real numbers $\R$.
Thus $S \subseteq \R$.
By definition:
:$\forall n \in \N: n \ge 0$
and so:
:$\forall n \in S: n \ge 0$
Hence $0$ is a lower bound of $S$.
This establishes the fact that $S... | Let $S$ be a [[Definition:Set|set]].
Then there exists a [[Definition:Total Ordering|total ordering]] on $S$. | Let $S$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Natural Numbers|set of natural numbers]] $\N$.
We take as [[Definition:Axiom|axiomatic]] that $\N$ is itself a [[Definition:Subset|subset]] of the [[Definition:Real Number|set of real numbers]] $\R$.
Thus $S \subseteq... | Well-Ordering Principle/Proof by Restriction of Real Numbers | https://proofwiki.org/wiki/Ordering_Principle | https://proofwiki.org/wiki/Well-Ordering_Principle/Proof_by_Restriction_of_Real_Numbers | [
"Set Theory",
"Total Orderings",
"Order Theory",
"Named Theorems",
"Ordering Principle"
] | [
"Definition:Set",
"Definition:Total Ordering"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Natural Numbers",
"Definition:Axiom",
"Definition:Subset",
"Definition:Real Number",
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Bounded Below Set/Real Numbers",
"Continuum Property",
"Definition:Infimum of Set/Real Numbers... |
proofwiki-6483 | Ordering Principle | Let $S$ be a set.
Then there exists a total ordering on $S$. | Consider the natural numbers $\N$ defined as the naturally ordered semigroup $\struct {S, \circ, \preceq}$.
From its definition, $\struct {S, \circ, \preceq}$ is well-ordered by $\preceq$.
The result follows.
As $\N_{\ne 0} = \N \setminus \set 0$, by Set Difference is Subset $\N_{\ne 0} \subseteq \N$.
As $\N$ is well-o... | Let $S$ be a [[Definition:Set|set]].
Then there exists a [[Definition:Total Ordering|total ordering]] on $S$. | Consider the [[Definition:Natural Numbers|natural numbers]] $\N$ defined as the [[Definition:Naturally Ordered Semigroup|naturally ordered semigroup]] $\struct {S, \circ, \preceq}$.
From its definition, $\struct {S, \circ, \preceq}$ is [[Definition:Well-Ordered Set|well-ordered]] by $\preceq$.
The result follows.
A... | Well-Ordering Principle/Proof using Naturally Ordered Semigroup | https://proofwiki.org/wiki/Ordering_Principle | https://proofwiki.org/wiki/Well-Ordering_Principle/Proof_using_Naturally_Ordered_Semigroup | [
"Set Theory",
"Total Orderings",
"Order Theory",
"Named Theorems",
"Ordering Principle"
] | [
"Definition:Set",
"Definition:Total Ordering"
] | [
"Definition:Natural Numbers",
"Definition:Naturally Ordered Semigroup",
"Definition:Well-Ordered Set",
"Set Difference is Subset",
"Definition:Well-Ordered Set",
"Definition:Smallest Element"
] |
proofwiki-6484 | Ordering Principle | Let $S$ be a set.
Then there exists a total ordering on $S$. | From Von Neumann Construction of Natural Numbers is Minimally Inductive, $\omega$ is a minimally inductive class under the successor mapping.
From Successor Mapping on Natural Numbers is Progressing, this successor mapping is a progressing mapping.
The result is a direct application of Minimally Inductive Class under P... | Let $S$ be a [[Definition:Set|set]].
Then there exists a [[Definition:Total Ordering|total ordering]] on $S$. | From [[Von Neumann Construction of Natural Numbers is Minimally Inductive]], $\omega$ is a [[Definition:Minimally Inductive Class under General Mapping|minimally inductive class]] under the [[Definition:Successor Mapping on Von Neumann Construction|successor mapping]].
From [[Successor Mapping on Natural Numbers is Pr... | Well-Ordering Principle/Proof using Von Neumann Construction | https://proofwiki.org/wiki/Ordering_Principle | https://proofwiki.org/wiki/Well-Ordering_Principle/Proof_using_Von_Neumann_Construction | [
"Set Theory",
"Total Orderings",
"Order Theory",
"Named Theorems",
"Ordering Principle"
] | [
"Definition:Set",
"Definition:Total Ordering"
] | [
"Von Neumann Construction of Natural Numbers is Minimally Inductive",
"Definition:Minimally Inductive Class under General Mapping",
"Definition:Natural Numbers/Von Neumann Construction/Successor Mapping",
"Successor Mapping on Natural Numbers is Progressing",
"Definition:Natural Numbers/Von Neumann Construc... |
proofwiki-6485 | Number Smaller than Lebesgue Number is also Lebesgue Number | Let $M = \struct {A, d}$ be a metric space.
Let $\epsilon \in \R_{>0}$ be a Lebesgue number for $M$.
Let $\epsilon' \in \R_{>0}: \epsilon' < \epsilon$.
Then $\epsilon'$ is also a Lebesgue number for $M$. | By hypothesis, let $\epsilon \in \R_{>0}$ be a Lebesgue number for $M$.
Then by definition:
:$\forall x \in A: \exists \map U x \in \UU: \map {B_\epsilon} x \subseteq \map U x$
where $\map {B_\epsilon} x$ is the open $\epsilon$-ball of $x$ in $M$.
Let $\epsilon' \in \R_{>0}: \epsilon' < \epsilon$.
Let $y \in \map {B_{\... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\epsilon \in \R_{>0}$ be a [[Definition:Lebesgue Number|Lebesgue number]] for $M$.
Let $\epsilon' \in \R_{>0}: \epsilon' < \epsilon$.
Then $\epsilon'$ is also a [[Definition:Lebesgue Number|Lebesgue number]] for $M$. | [[Definition:By Hypothesis|By hypothesis]], let $\epsilon \in \R_{>0}$ be a [[Definition:Lebesgue Number|Lebesgue number]] for $M$.
Then by definition:
:$\forall x \in A: \exists \map U x \in \UU: \map {B_\epsilon} x \subseteq \map U x$
where $\map {B_\epsilon} x$ is the [[Definition:Open Ball of Metric Space|open $\e... | Number Smaller than Lebesgue Number is also Lebesgue Number | https://proofwiki.org/wiki/Number_Smaller_than_Lebesgue_Number_is_also_Lebesgue_Number | https://proofwiki.org/wiki/Number_Smaller_than_Lebesgue_Number_is_also_Lebesgue_Number | [
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Lebesgue Number",
"Definition:Lebesgue Number"
] | [
"Definition:By Hypothesis",
"Definition:Lebesgue Number",
"Definition:Open Ball",
"Definition:Lebesgue Number"
] |
proofwiki-6486 | Open Cover may not have Lebesgue Number | Let $M = \struct {A, d}$ be a metric space.
Let $\CC$ be an open cover of $M$.
Then it may not necessarily be the case that $\CC$ has a Lebesgue number. | Let $M := \openint 0 1 \subseteq \R$ under the Euclidean metric.
Let $\CC := \set {\openint {\dfrac 1 n} 1: n \ge 2}$.
For any $\epsilon \in \R_{>0}$, take $n > \dfrac 1 \epsilon$ and $x = \dfrac 1 n$.
There is no $\openint {\dfrac 1 m} 1$ such that $\map {B_\epsilon} x \subseteq \openint {\dfrac 1 m} 1$, since $\map {... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $\CC$ be an [[Definition:Open Cover|open cover]] of $M$.
Then it may not necessarily be the case that $\CC$ has a [[Definition:Lebesgue Number|Lebesgue number]]. | Let $M := \openint 0 1 \subseteq \R$ under the [[Definition:Euclidean Metric on Real Number Line|Euclidean metric]].
Let $\CC := \set {\openint {\dfrac 1 n} 1: n \ge 2}$.
For any $\epsilon \in \R_{>0}$, take $n > \dfrac 1 \epsilon$ and $x = \dfrac 1 n$.
There is no $\openint {\dfrac 1 m} 1$ such that $\map {B_\epsil... | Open Cover may not have Lebesgue Number | https://proofwiki.org/wiki/Open_Cover_may_not_have_Lebesgue_Number | https://proofwiki.org/wiki/Open_Cover_may_not_have_Lebesgue_Number | [
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Open Cover",
"Definition:Lebesgue Number"
] | [
"Definition:Euclidean Metric/Real Number Line"
] |
proofwiki-6487 | Burnside's Theorem | Let $G$ be a finite group.
Let the order of $G$ be $p^m q^n$ where:
:$p, q$ are prime
:$m, n \in \N$
Then $G$ is solvable. | {{ProofWanted}}
{{Namedfor|William Burnside|cat = Burnside}} | Let $G$ be a [[Definition:Finite Group|finite group]].
Let the [[Definition:Order of Group|order]] of $G$ be $p^m q^n$ where:
:$p, q$ are [[Definition:Prime Number|prime]]
:$m, n \in \N$
Then $G$ is [[Definition:Solvable Group|solvable]]. | {{ProofWanted}}
{{Namedfor|William Burnside|cat = Burnside}} | Burnside's Theorem | https://proofwiki.org/wiki/Burnside's_Theorem | https://proofwiki.org/wiki/Burnside's_Theorem | [
"Solvable Groups",
"Finite Groups"
] | [
"Definition:Finite Group",
"Definition:Order of Structure",
"Definition:Prime Number",
"Definition:Solvable Group"
] | [] |
proofwiki-6488 | Constant Function is Uniformly Continuous/Metric Space | Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.
Let $f_c: A_1 \to A_2$ be the constant mapping from $A_1$ to $A_2$:
:$\exists c \in A_2: \forall a \in A_1: f_c \left({a}\right) = c$
That is, every point in $A_1$ maps to the same point $c$ in $A_2$.
Then $f_c$ is uniformly conti... | Let $f_c: A_1 \to A_2$ be the constant mapping between two metric spaces $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$.
Let $\epsilon \in \R: \epsilon > 0$.
Pick any $\delta \in \R$ such that $\delta > 0$.
Let $x, y \in A_1$ such that $d_1 \left({x, y}\right) < \delta$.
Now we have:
:$f_c \left({x... | Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be [[Definition:Metric Space|metric spaces]].
Let $f_c: A_1 \to A_2$ be the [[Definition:Constant Mapping|constant mapping]] from $A_1$ to $A_2$:
:$\exists c \in A_2: \forall a \in A_1: f_c \left({a}\right) = c$
That is, every [[Definition:Eleme... | Let $f_c: A_1 \to A_2$ be the [[Definition:Constant Mapping|constant mapping]] between two [[Definition:Metric Space|metric spaces]] $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$.
Let $\epsilon \in \R: \epsilon > 0$.
Pick any $\delta \in \R$ such that $\delta > 0$.
Let $x, y \in A_1$ such that ... | Constant Function is Uniformly Continuous/Metric Space | https://proofwiki.org/wiki/Constant_Function_is_Uniformly_Continuous/Metric_Space | https://proofwiki.org/wiki/Constant_Function_is_Uniformly_Continuous/Metric_Space | [
"Constant Mappings",
"Metric Spaces",
"Uniformly Continuous Mappings"
] | [
"Definition:Metric Space",
"Definition:Constant Mapping",
"Definition:Element",
"Definition:Element",
"Definition:Uniform Continuity/Metric Space"
] | [
"Definition:Constant Mapping",
"Definition:Metric Space",
"Definition:Metric Space/Metric",
"Definition:Uniform Continuity/Metric Space",
"Definition:Uniform Continuity/Metric Space",
"Category:Constant Mappings",
"Category:Metric Spaces",
"Category:Uniformly Continuous Mappings"
] |
proofwiki-6489 | Constant Function is Uniformly Continuous/Real Function | Let $f_c: \R \to \R$ be the constant mapping:
:$\exists c \in \R: \forall a \in \R: \map {f_c} a = c$
Then $f_c$ is uniformly continuous on $\R$. | Follows directly from:
:Constant Function is Uniformly Continuous: Metric Space
:Real Number Line is Metric Space.
{{qed}} | Let $f_c: \R \to \R$ be the [[Definition:Constant Mapping|constant mapping]]:
:$\exists c \in \R: \forall a \in \R: \map {f_c} a = c$
Then $f_c$ is [[Definition:Uniformly Continuous Real Function|uniformly continuous on $\R$]]. | Follows directly from:
:[[Constant Function is Uniformly Continuous/Metric Space|Constant Function is Uniformly Continuous: Metric Space]]
:[[Real Number Line is Metric Space]].
{{qed}} | Constant Function is Uniformly Continuous/Real Function | https://proofwiki.org/wiki/Constant_Function_is_Uniformly_Continuous/Real_Function | https://proofwiki.org/wiki/Constant_Function_is_Uniformly_Continuous/Real_Function | [
"Constant Mappings",
"Real Analysis",
"Uniformly Continuous Real Functions"
] | [
"Definition:Constant Mapping",
"Definition:Uniform Continuity/Real Function"
] | [
"Constant Function is Uniformly Continuous/Metric Space",
"Real Number Line is Metric Space"
] |
proofwiki-6490 | Linear Combination of Complex Integrals | Let $\closedint a b$ be a closed real interval.
Let $f, g: \closedint a b \to \C$ be complex Riemann integrable functions over $\closedint a b$.
Let $\lambda, \mu \in \C$ be complex constants.
Then:
:$\ds \int_a^b \paren {\lambda \map f t + \mu \map g t} \rd t = \lambda \int_a^b \map f t \rd t + \mu \int_a^b \map g t \... | First, we prove the result for addition only without multiplying by $\lambda, \mu$:
{{begin-eqn}}
{{eqn | l = \int_a^b \map f t + \map g t \rd t
| r = \int_a^b \map \Re {\map f t + \map g t} \rd t + i \int_a^b \map \Im {\map f t + \map g t} \rd t
| c = {{Defof|Complex Riemann Integral}}
}}
{{eqn | r = \int_... | Let $\closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $f, g: \closedint a b \to \C$ be [[Definition:Complex Riemann Integral|complex Riemann integrable]] [[Definition:Real Function|functions]] over $\closedint a b$.
Let $\lambda, \mu \in \C$ be [[Definition:Complex Number|complex]] [... | First, we prove the result for addition only without multiplying by $\lambda, \mu$:
{{begin-eqn}}
{{eqn | l = \int_a^b \map f t + \map g t \rd t
| r = \int_a^b \map \Re {\map f t + \map g t} \rd t + i \int_a^b \map \Im {\map f t + \map g t} \rd t
| c = {{Defof|Complex Riemann Integral}}
}}
{{eqn | r = \int... | Linear Combination of Complex Integrals | https://proofwiki.org/wiki/Linear_Combination_of_Complex_Integrals | https://proofwiki.org/wiki/Linear_Combination_of_Complex_Integrals | [
"Complex Analysis"
] | [
"Definition:Real Interval/Closed",
"Definition:Integrable Function/Complex",
"Definition:Real Function",
"Definition:Complex Number",
"Definition:Constant"
] | [
"Addition of Real and Imaginary Parts",
"Linear Combination of Integrals/Definite",
"Definition:Complex Number",
"Addition of Real and Imaginary Parts",
"Multiplication of Real and Imaginary Parts",
"Linear Combination of Integrals/Definite"
] |
proofwiki-6491 | Pointwise Difference of Simple Functions is Simple Function | Let $\struct {X, \Sigma}$ be a measurable space.
Let $f,g : X \to \R$ be simple functions.
Let $f - g: X \to \R$ be the pointwise difference of $f$ and $g$:
:$\forall x \in X: \map {\paren {f - g} } x := \map f x - \map g x$
Then $f - g$ is also a simple function. | By Scalar Multiple of Simple Function is Simple Function, $-g = -1 \cdot g$ is a simple function.
By Pointwise Sum of Simple Functions is Simple Function, so is $f - g$.
{{qed}}
Category:Simple Functions
7evs65fh65472uutz4nfnefrd9atbrp | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $f,g : X \to \R$ be [[Definition:Simple Function|simple functions]].
Let $f - g: X \to \R$ be the [[Definition:Pointwise Subtraction|pointwise difference]] of $f$ and $g$:
:$\forall x \in X: \map {\paren {f - g} } x := \map f x - \ma... | By [[Scalar Multiple of Simple Function is Simple Function]], $-g = -1 \cdot g$ is a [[Definition:Simple Function|simple function]].
By [[Pointwise Sum of Simple Functions is Simple Function]], so is $f - g$.
{{qed}}
[[Category:Simple Functions]]
7evs65fh65472uutz4nfnefrd9atbrp | Pointwise Difference of Simple Functions is Simple Function | https://proofwiki.org/wiki/Pointwise_Difference_of_Simple_Functions_is_Simple_Function | https://proofwiki.org/wiki/Pointwise_Difference_of_Simple_Functions_is_Simple_Function | [
"Simple Functions"
] | [
"Definition:Measurable Space",
"Definition:Simple Function",
"Definition:Pointwise Subtraction",
"Definition:Simple Function"
] | [
"Scalar Multiple of Simple Function is Simple Function",
"Definition:Simple Function",
"Pointwise Sum of Simple Functions is Simple Function",
"Category:Simple Functions"
] |
proofwiki-6492 | Modulus and Argument of Complex Exponential | Let $z \in \C$ be a complex number.
Let $\hointr a {a + 2 \pi}$ be a half open interval of length $2 \pi$.
Let $r \in \hointr 0 {+\infty}$ and $\theta \in \hointr a {a + 2 \pi}$.
Then:
:$r = \cmod z$ and $\theta = \map \arg z$
{{iff}}:
:$z = r e^{i \theta}$
where:
:$\cmod z$ denotes the modulus of $z$
:$\map \arg z$ de... | === Necessary condition ===
Let $r = \cmod z$.
If $z = 0$, we have:
:$z = 0e^{i \theta} = re^{i \theta}$
Suppose $z \ne 0$ and $\theta = \map \arg z$.
By definition of argument, the following two equations hold:
:$(1): \quad \dfrac {\map \Re z} r = \cos \theta$
:$(2): \quad \dfrac {\map \Im z} r = \sin \theta$
where:
:... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]].
Let $\hointr a {a + 2 \pi}$ be a [[Definition:Half-Open Real Interval|half open interval]] of [[Definition:Length of Real Interval|length]] $2 \pi$.
Let $r \in \hointr 0 {+\infty}$ and $\theta \in \hointr a {a + 2 \pi}$.
Then:
:$r = \cmod z$ and $\th... | === Necessary condition ===
Let $r = \cmod z$.
If $z = 0$, we have:
:$z = 0e^{i \theta} = re^{i \theta}$
Suppose $z \ne 0$ and $\theta = \map \arg z$.
By definition of [[Definition:Argument of Complex Number|argument]], the following two equations hold:
:$(1): \quad \dfrac {\map \Re z} r = \cos \theta$
:$(2): \qua... | Modulus and Argument of Complex Exponential | https://proofwiki.org/wiki/Modulus_and_Argument_of_Complex_Exponential | https://proofwiki.org/wiki/Modulus_and_Argument_of_Complex_Exponential | [
"Complex Analysis",
"Complex Modulus"
] | [
"Definition:Complex Number",
"Definition:Real Interval/Half-Open",
"Definition:Real Interval/Length",
"Definition:Complex Modulus",
"Definition:Argument of Complex Number",
"Definition:Exponential Function/Complex"
] | [
"Definition:Argument of Complex Number",
"Definition:Complex Number/Real Part",
"Definition:Complex Number/Imaginary Part",
"Euler's Formula",
"Definition:Argument of Complex Number"
] |
proofwiki-6493 | Bases of Vector Space have Equal Cardinality | Let $R$ be a division ring.
Let $V$ be a vector space over $R$.
Let $X$ and $Y$ be bases of $V$.
Then $X$ and $Y$ are equivalent.
That is, $X$ and $Y$ have the same cardinality. | We will first prove that there is an injection from $X$ to $Y$.
Let $x \in X$.
By Expression of Vector as Linear Combination from Basis is Unique: General Result, there is a unique finite subset $C_x$ of $R \times Y$ such that:
:$\ds x = \sum_{\tuple {r, v} \mathop \in C_x} r \cdot v$ and
:$\forall \tuple {r, v} \in C_... | Let $R$ be a [[Definition:Division Ring|division ring]].
Let $V$ be a [[Definition:Vector Space|vector space]] over $R$.
Let $X$ and $Y$ be [[Definition:Basis of Vector Space|bases]] of $V$.
Then $X$ and $Y$ are [[Definition:Set Equivalence|equivalent]].
That is, $X$ and $Y$ have the same [[Definition:Cardinality|... | We will first prove that there is an [[Definition:Injection|injection]] from $X$ to $Y$.
Let $x \in X$.
By [[Expression of Vector as Linear Combination from Basis is Unique/General Result|Expression of Vector as Linear Combination from Basis is Unique: General Result]], there is a unique [[Definition:Finite Set|fini... | Bases of Vector Space have Equal Cardinality | https://proofwiki.org/wiki/Bases_of_Vector_Space_have_Equal_Cardinality | https://proofwiki.org/wiki/Bases_of_Vector_Space_have_Equal_Cardinality | [
"Bases of Vector Spaces"
] | [
"Definition:Division Ring",
"Definition:Vector Space",
"Definition:Basis of Vector Space",
"Definition:Set Equivalence",
"Definition:Cardinality"
] | [
"Definition:Injection",
"Expression of Vector as Linear Combination from Basis is Unique/General Result",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Basis of Vector Space",
"Definition:Linearly Independent/Set",
"Subset of Linearly Indepen... |
proofwiki-6494 | Equivalence of Definitions of Symmetric Difference/(1) iff (2) | Let $S$ and $T$ be sets.
{{TFAENocat|def = Symmetric Difference|view = symmetric difference $S \symdif T$ between $S$ and $T$}} | {{begin-eqn}}
{{eqn | l = S \symdif T
| r = \paren {S \setminus T} \cup \paren {T \setminus S}
| c = {{Defof|Symmetric Difference|index = 1}}
}}
{{eqn | r = \paren {\paren {S \cup T} \setminus T} \cup \paren {\paren {S \cup T} \setminus S}
| c = Set Difference with Union is Set Difference
}}
{{eqn | r... | Let $S$ and $T$ be [[Definition:Set|sets]].
{{TFAENocat|def = Symmetric Difference|view = symmetric difference $S \symdif T$ between $S$ and $T$}} | {{begin-eqn}}
{{eqn | l = S \symdif T
| r = \paren {S \setminus T} \cup \paren {T \setminus S}
| c = {{Defof|Symmetric Difference|index = 1}}
}}
{{eqn | r = \paren {\paren {S \cup T} \setminus T} \cup \paren {\paren {S \cup T} \setminus S}
| c = [[Set Difference with Union is Set Difference]]
}}
{{eqn... | Equivalence of Definitions of Symmetric Difference/(1) iff (2) | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Symmetric_Difference/(1)_iff_(2) | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Symmetric_Difference/(1)_iff_(2) | [
"Equivalence of Definitions of Symmetric Difference"
] | [
"Definition:Set"
] | [
"Set Difference with Union is Set Difference",
"De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection",
"Intersection is Commutative"
] |
proofwiki-6495 | Equivalence of Definitions of Symmetric Difference/(1) iff (3) | Let $S$ and $T$ be sets.
{{TFAENocat|def = Symmetric Difference|view = symmetric difference $S \symdif T$ between $S$ and $T$}} | {{begin-eqn}}
{{eqn | l = S \symdif T
| r = \paren {S \setminus T} \cup \paren {T \setminus S}
| c = {{Defof|Symmetric Difference|index = 1}}
}}
{{eqn | r = \paren {S \cap \overline T} \cup \paren {\overline S \cap T}
| c = Set Difference as Intersection with Complement
}}
{{end-eqn}} | Let $S$ and $T$ be [[Definition:Set|sets]].
{{TFAENocat|def = Symmetric Difference|view = symmetric difference $S \symdif T$ between $S$ and $T$}} | {{begin-eqn}}
{{eqn | l = S \symdif T
| r = \paren {S \setminus T} \cup \paren {T \setminus S}
| c = {{Defof|Symmetric Difference|index = 1}}
}}
{{eqn | r = \paren {S \cap \overline T} \cup \paren {\overline S \cap T}
| c = [[Set Difference as Intersection with Complement]]
}}
{{end-eqn}} | Equivalence of Definitions of Symmetric Difference/(1) iff (3) | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Symmetric_Difference/(1)_iff_(3) | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Symmetric_Difference/(1)_iff_(3) | [
"Equivalence of Definitions of Symmetric Difference"
] | [
"Definition:Set"
] | [
"Set Difference as Intersection with Complement"
] |
proofwiki-6496 | Equivalence of Definitions of Symmetric Difference/(2) iff (4) | Let $S$ and $T$ be sets.
{{TFAENocat|def = Symmetric Difference|view = symmetric difference $S \symdif T$ between $S$ and $T$}} | {{begin-eqn}}
{{eqn | l = S \symdif T
| r = \paren {S \cup T} \setminus \paren {S \cap T}
| c = {{Defof|Symmetric Difference|index = 2}}
}}
{{eqn | r = \paren {S \cup T} \cap \paren {\overline {S \cap T} }
| c = Set Difference as Intersection with Complement
}}
{{eqn | r = \paren {S \cup T} \cap \pare... | Let $S$ and $T$ be [[Definition:Set|sets]].
{{TFAENocat|def = Symmetric Difference|view = symmetric difference $S \symdif T$ between $S$ and $T$}} | {{begin-eqn}}
{{eqn | l = S \symdif T
| r = \paren {S \cup T} \setminus \paren {S \cap T}
| c = {{Defof|Symmetric Difference|index = 2}}
}}
{{eqn | r = \paren {S \cup T} \cap \paren {\overline {S \cap T} }
| c = [[Set Difference as Intersection with Complement]]
}}
{{eqn | r = \paren {S \cup T} \cap \... | Equivalence of Definitions of Symmetric Difference/(2) iff (4) | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Symmetric_Difference/(2)_iff_(4) | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Symmetric_Difference/(2)_iff_(4) | [
"Equivalence of Definitions of Symmetric Difference"
] | [
"Definition:Set"
] | [
"Set Difference as Intersection with Complement",
"De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection"
] |
proofwiki-6497 | Subset of Countable Set is Countable | A subset of a countable set is countable. | Let $S$ be a countable set.
Let $T \subseteq S$.
By definition, there exists an injection $f: S \to \N$.
Let $i: T \to S$ be the inclusion mapping.
We have that $i$ is an injection.
Because the composite of injections is an injection, it follows that $f \circ i: T \to \N$ is an injection.
Hence, $T$ is countable.
{{qed... | A [[Definition:Subset|subset]] of a [[Definition:Countable Set|countable set]] is [[Definition:Countable Set|countable]]. | Let $S$ be a [[Definition:Countable Set|countable set]].
Let $T \subseteq S$.
By definition, there exists an [[Definition:Injection|injection]] $f: S \to \N$.
Let $i: T \to S$ be the [[Definition:Inclusion Mapping|inclusion mapping]].
We have that [[Inclusion Mapping is Injection|$i$ is an injection]].
Because th... | Subset of Countable Set is Countable | https://proofwiki.org/wiki/Subset_of_Countable_Set_is_Countable | https://proofwiki.org/wiki/Subset_of_Countable_Set_is_Countable | [
"Countable Sets",
"Subsets"
] | [
"Definition:Subset",
"Definition:Countable Set",
"Definition:Countable Set"
] | [
"Definition:Countable Set",
"Definition:Injection",
"Definition:Inclusion Mapping",
"Inclusion Mapping is Injection",
"Composite of Injections is Injection",
"Definition:Injection",
"Definition:Countable Set"
] |
proofwiki-6498 | Euler's Sine Identity | :$\sin z = \dfrac {e^{i z} - e^{-i z} } {2 i}$ | Recall the definition of the sine function:
{{begin-eqn}}
{{eqn | l = \sin z
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + 1}!}
| c =
}}
{{eqn | r = z - \frac {z^3} {3!} + \frac {z^5} {5!} - \frac {z^7} {7!} + \cdots + \paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + ... | :$\sin z = \dfrac {e^{i z} - e^{-i z} } {2 i}$ | Recall the definition of the [[Definition:Complex Sine Function|sine function]]:
{{begin-eqn}}
{{eqn | l = \sin z
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1} } {\paren {2 n + 1}!}
| c =
}}
{{eqn | r = z - \frac {z^3} {3!} + \frac {z^5} {5!} - \frac {z^7} {7!} + \cdots + \paren {-1}^... | Euler's Sine Identity/Proof 1 | https://proofwiki.org/wiki/Euler's_Sine_Identity | https://proofwiki.org/wiki/Euler's_Sine_Identity/Proof_1 | [
"Euler's Sine Identity",
"Euler's Identities",
"Sine Function"
] | [] | [
"Definition:Sine/Complex Function",
"Definition:Exponential Function/Complex/Power Series Expansion",
"Definition:Even Integer",
"Definition:Odd Integer"
] |
proofwiki-6499 | Euler's Sine Identity | :$\sin z = \dfrac {e^{i z} - e^{-i z} } {2 i}$ | Recall Euler's Formula:
:$e^{i z} = \cos z + i \sin z$
Then, starting from the {{RHS}}:
{{begin-eqn}}
{{eqn | l = \frac {e^{i z} - e^{-i z} } {2 i}
| r = \frac {\paren {\cos z + i \sin z} - \paren {\map \cos {-z} + i \map \sin {-z} } } {2 i}
}}
{{eqn | r = \frac {\paren {\cos z + i \sin z - \cos z - i \map \sin {... | :$\sin z = \dfrac {e^{i z} - e^{-i z} } {2 i}$ | Recall [[Euler's Formula]]:
:$e^{i z} = \cos z + i \sin z$
Then, starting from the {{RHS}}:
{{begin-eqn}}
{{eqn | l = \frac {e^{i z} - e^{-i z} } {2 i}
| r = \frac {\paren {\cos z + i \sin z} - \paren {\map \cos {-z} + i \map \sin {-z} } } {2 i}
}}
{{eqn | r = \frac {\paren {\cos z + i \sin z - \cos z - i \ma... | Euler's Sine Identity/Proof 2 | https://proofwiki.org/wiki/Euler's_Sine_Identity | https://proofwiki.org/wiki/Euler's_Sine_Identity/Proof_2 | [
"Euler's Sine Identity",
"Euler's Identities",
"Sine Function"
] | [] | [
"Euler's Formula",
"Cosine Function is Even",
"Sine Function is Odd"
] |
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