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-23200 | Euler's Continued Fraction Formula/Corollary 2 | :$A - B + C - D + E - \cdots = \cfrac A {1 + \cfrac {B} {A - B + \cfrac {A C} {B - C + \cfrac {B D} {C - D + \cfrac {C E} {D - E + \cfrac {\ddots} {\ddots} } } } } }$ | From Euler's Continued Fraction Formula, we have:
{{begin-eqn}}
{{eqn | l = a_0 + a_0 a_1 + a_0 a_1 a_2 + a_0 a_1 a_2 a_3 + \cdots + a_0 a_1 a_2 a_3 \cdots a_n
| r = a_0 \paren {1 + a_1 \paren {1 + a_2 \paren { 1 + a_3 \paren {\cdots + a_n } } } }
| c =
}}
{{eqn | r = \cfrac {a_0} {1 - \cfrac {a_1} {1 + a_... | :$A - B + C - D + E - \cdots = \cfrac A {1 + \cfrac {B} {A - B + \cfrac {A C} {B - C + \cfrac {B D} {C - D + \cfrac {C E} {D - E + \cfrac {\ddots} {\ddots} } } } } }$ | From [[Euler's Continued Fraction Formula]], we have:
{{begin-eqn}}
{{eqn | l = a_0 + a_0 a_1 + a_0 a_1 a_2 + a_0 a_1 a_2 a_3 + \cdots + a_0 a_1 a_2 a_3 \cdots a_n
| r = a_0 \paren {1 + a_1 \paren {1 + a_2 \paren { 1 + a_3 \paren {\cdots + a_n } } } }
| c =
}}
{{eqn | r = \cfrac {a_0} {1 - \cfrac {a_1} {1... | Euler's Continued Fraction Formula/Corollary 2 | https://proofwiki.org/wiki/Euler's_Continued_Fraction_Formula/Corollary_2 | https://proofwiki.org/wiki/Euler's_Continued_Fraction_Formula/Corollary_2 | [
"Euler's Continued Fraction Formula"
] | [] | [
"Euler's Continued Fraction Formula"
] |
proofwiki-23201 | Continued Fraction for Pi plus Three | :$\pi + 3 = 6 + \cfrac {1^2} {6 + \cfrac {3^2} {6 + \cfrac {5^2} {6 + \cfrac {7^2} {6 + \cfrac {9^2} {6 + \cfrac \ddots \ddots} } } } }$ | Let:
{{begin-eqn}}
{{eqn | l = S
| r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } {\paren {2 n} \paren {2 n + 1} \paren {2 n + 2} }
| c =
}}
{{eqn | n = 1
| r = \frac 1 {2 \times 3 \times 4} - \frac 1 {4 \times 5 \times 6} + \frac 1 {6 \times 7 \times 8} - \frac 1 {8 \times 9 \times 10... | :$\pi + 3 = 6 + \cfrac {1^2} {6 + \cfrac {3^2} {6 + \cfrac {5^2} {6 + \cfrac {7^2} {6 + \cfrac {9^2} {6 + \cfrac \ddots \ddots} } } } }$ | Let:
{{begin-eqn}}
{{eqn | l = S
| r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } {\paren {2 n} \paren {2 n + 1} \paren {2 n + 2} }
| c =
}}
{{eqn | n = 1
| r = \frac 1 {2 \times 3 \times 4} - \frac 1 {4 \times 5 \times 6} + \frac 1 {6 \times 7 \times 8} - \frac 1 {8 \times 9 \times 10... | Continued Fraction for Pi plus Three | https://proofwiki.org/wiki/Continued_Fraction_for_Pi_plus_Three | https://proofwiki.org/wiki/Continued_Fraction_for_Pi_plus_Three | [
"Examples of Continued Fractions",
"Examples of Euler's Continued Fraction Formula",
"Pi"
] | [] | [
"Partial Fractions Expansion/Examples/1 over 2x(2x+1)(2x+2)",
"Leibniz's Formula for Pi"
] |
proofwiki-23202 | Coefficients of Coordinates in General Equation of Plane Form Vector Perpendicular to Plane | Let $P$ be a plane that is described by the general equation:
:$\alpha_1 x + \alpha_2 y + \alpha_3 z = \gamma$
Let $\mathbf i$, $\mathbf j$ and $\mathbf k$ denote the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis, respectively.
Then the vector:
:$\bsalpha = \alpha_1 \mathbf i + \alpha_2... | Let $A$ and $B$ be two points in $P$.
Let $\mathbf a$ and $\mathbf b$ be two vectors from the origin to $A$ and $B$, respectively.
Let $A$ and $B$ have the following coordinates:
:$A = \tuple {x_a, y_a, z_a}$
:$B = \tuple {x_b, y_b, z_b}$
Since the points are in $P$, they satisfy
:$\alpha_1 x_a + \alpha_2 y_a + \alpha_... | Let $P$ be a [[Definition:Plane|plane]] that is described by the [[General Equation of Plane|general equation]]:
:$\alpha_1 x + \alpha_2 y + \alpha_3 z = \gamma$
Let $\mathbf i$, $\mathbf j$ and $\mathbf k$ denote the [[Definition:Unit Vector|unit vectors]] in the [[Definition:Positive Direction|positive directions]]... | Let $A$ and $B$ be two [[Definition:Point|points]] in $P$.
Let $\mathbf a$ and $\mathbf b$ be two [[Definition:Space Vector|vectors]] from the [[Definition:Origin|origin]] to $A$ and $B$, respectively.
Let $A$ and $B$ have the following [[Definition:Coordinates|coordinates]]:
:$A = \tuple {x_a, y_a, z_a}$
:$B = \tupl... | Coefficients of Coordinates in General Equation of Plane Form Vector Perpendicular to Plane | https://proofwiki.org/wiki/Coefficients_of_Coordinates_in_General_Equation_of_Plane_Form_Vector_Perpendicular_to_Plane | https://proofwiki.org/wiki/Coefficients_of_Coordinates_in_General_Equation_of_Plane_Form_Vector_Perpendicular_to_Plane | [
"Planes",
"Equations of Planes",
"Perpendiculars"
] | [
"Definition:Plane Surface",
"Equation of Plane/General Equation",
"Definition:Unit Vector",
"Definition:Axis/Positive Direction",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis",
"Definition:Axis/Z-Axis",
"Definition:Vector/Real Euclidean Space/Space Vector",
"Definition:Right Angle/Perpendicular... | [
"Definition:Point",
"Definition:Vector/Real Euclidean Space/Space Vector",
"Definition:Coordinate System/Origin",
"Definition:Coordinate System/Coordinate",
"Definition:Point",
"Definition:Point",
"Definition:Arbitrary",
"Dot Product of Perpendicular Vectors",
"Definition:Right Angle/Perpendicular/P... |
proofwiki-23203 | Square Matrix has Linearly Dependent Rows iff Determinant is Zero | Let $M$ be an $n \times n$ square matrix.
Then:
:the rows of $M$ are linearly dependent
{{iff}}:
:$\map \det M = 0$ | === Sufficient Condition ===
Let the rows of $M$ be linearly dependent.
Then there exists a set of coefficients $\set {\lambda_1, \lambda_2, \ldots, \lambda_n}$ such that:
:$\ds \sum_{i \mathop = 1}^n \lambda_i r_i = \mathbf 0$
where:
:$r_i$ is the $i$th row of $M$
:$\exists i \in \set {1, 2, \ldots, n} : \lambda_i \ne... | Let $M$ be an $n \times n$ [[Definition:Square Matrix|square matrix]].
Then:
:the [[Definition:Row of Matrix|rows]] of $M$ are [[Definition:Linearly Dependent Set of Real Vectors|linearly dependent]]
{{iff}}:
:$\map \det M = 0$ | === Sufficient Condition ===
Let the [[Definition:Row of Matrix|rows]] of $M$ be [[Definition:Linearly Dependent Set of Real Vectors|linearly dependent]].
Then there exists a [[Definition:Set|set]] of [[Definition:Coefficient|coefficients]] $\set {\lambda_1, \lambda_2, \ldots, \lambda_n}$ such that:
:$\ds \sum_{i \ma... | Square Matrix has Linearly Dependent Rows iff Determinant is Zero | https://proofwiki.org/wiki/Square_Matrix_has_Linearly_Dependent_Rows_iff_Determinant_is_Zero | https://proofwiki.org/wiki/Square_Matrix_has_Linearly_Dependent_Rows_iff_Determinant_is_Zero | [
"Determinants",
"Square Matrices",
"Linear Dependence"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Row",
"Definition:Linearly Dependent/Set/Real Vector Space"
] | [
"Definition:Matrix/Row",
"Definition:Linearly Dependent/Set/Real Vector Space",
"Definition:Set",
"Definition:Coefficient",
"Definition:Matrix/Row",
"Definition:Determinant/Matrix",
"Definition:Elementary Operation/Row",
"Definition:Matrix/Row",
"Definition:Matrix Scalar Product",
"Definition:Elem... |
proofwiki-23204 | Transpose Exchanges Rows and Columns | Let $A$ be an $m \times n$ matrix over a set.
Let $r_k$ be the $k$th row of $A$ for $1 \leq k \leq m$.
Let $A^\intercal$ be the transpose of $A$.
Let $c_k$ be the $k$th column of $A^\intercal$.
Then, $r_k = c_k$.
Similarly, the $k$th column of $A$ is the $k$th row of $A^\intercal$. | The $k$th row of $A$ is given by
:$r_k = \tuple {a_{k1}, a_{k2}, \ldots, a_{km} }$
where $a_{ij}$ is an matrix element of $A$.
The $k$th column of $A^\intercal$ is given by
:$c_k = \tuple {b_{1k}, b_{2k}, \ldots, b_{mk} }$
where $b_{ij}$ is an matrix element of $A^\intercal$.
According to the definition of transpose,
:... | Let $A$ be an $m \times n$ [[Definition:Matrix|matrix]] over a [[Definition:Set|set]].
Let $r_k$ be the $k$th [[Definition:Row of Matrix|row]] of $A$ for $1 \leq k \leq m$.
Let $A^\intercal$ be the [[Definition:Transpose of Matrix|transpose]] of $A$.
Let $c_k$ be the $k$th [[Definition:Column of Matrix|column]] of $... | The $k$th [[Definition:Row of Matrix|row]] of $A$ is given by
:$r_k = \tuple {a_{k1}, a_{k2}, \ldots, a_{km} }$
where $a_{ij}$ is an [[Definition:Element of Matrix|matrix element]] of $A$.
The $k$th [[Definition:Column of Matrix|column]] of $A^\intercal$ is given by
:$c_k = \tuple {b_{1k}, b_{2k}, \ldots, b_{mk} }$
w... | Transpose Exchanges Rows and Columns | https://proofwiki.org/wiki/Transpose_Exchanges_Rows_and_Columns | https://proofwiki.org/wiki/Transpose_Exchanges_Rows_and_Columns | [
"Transposes of Matrices",
"Matrices"
] | [
"Definition:Matrix",
"Definition:Set",
"Definition:Matrix/Row",
"Definition:Transpose of Matrix",
"Definition:Matrix/Column",
"Definition:Matrix/Column",
"Definition:Matrix/Row"
] | [
"Definition:Matrix/Row",
"Definition:Matrix/Element",
"Definition:Matrix/Column",
"Definition:Matrix/Element",
"Definition:Transpose of Matrix",
"Definition:Matrix/Column",
"Definition:Matrix/Row",
"Category:Transposes of Matrices",
"Category:Matrices"
] |
proofwiki-23205 | Square Matrix has Linearly Dependent Columns iff Determinant is Zero | Let $M$ be an $n \times n$ square matrix.
Then:
:the columns of $M$ are linearly dependent
{{iff}}:
:$\map \det M = 0$ | Let $M^\intercal$ be the transpose of $M$.
By Determinant of Transpose:
:$\map \det M = 0 \iff \map \det {M^\intercal} = 0$
By Square Matrix has Linearly Dependent Rows iff Determinant is Zero:
:$\map \det {M^\intercal} = 0 \iff$ Rows of $M^\intercal$ are linearly dependent
By Transpose Exchanges Rows and Columns:
:Row... | Let $M$ be an $n \times n$ [[Definition:Square Matrix|square matrix]].
Then:
:the [[Definition:Column of Matrix|columns]] of $M$ are [[Definition:Linearly Dependent Set of Real Vectors|linearly dependent]]
{{iff}}:
:$\map \det M = 0$ | Let $M^\intercal$ be the [[Definition:Transpose of Matrix|transpose]] of $M$.
By [[Determinant of Transpose]]:
:$\map \det M = 0 \iff \map \det {M^\intercal} = 0$
By [[Square Matrix has Linearly Dependent Rows iff Determinant is Zero]]:
:$\map \det {M^\intercal} = 0 \iff$ [[Definition:Row of Matrix|Rows]] of $M^\int... | Square Matrix has Linearly Dependent Columns iff Determinant is Zero | https://proofwiki.org/wiki/Square_Matrix_has_Linearly_Dependent_Columns_iff_Determinant_is_Zero | https://proofwiki.org/wiki/Square_Matrix_has_Linearly_Dependent_Columns_iff_Determinant_is_Zero | [
"Determinants",
"Square Matrices",
"Linear Dependence"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Column",
"Definition:Linearly Dependent/Set/Real Vector Space"
] | [
"Definition:Transpose of Matrix",
"Determinant of Transpose",
"Square Matrix has Linearly Dependent Rows iff Determinant is Zero",
"Definition:Matrix/Row",
"Definition:Linearly Dependent/Set/Real Vector Space",
"Transpose Exchanges Rows and Columns",
"Definition:Matrix/Row",
"Definition:Linearly Depen... |
proofwiki-23206 | Union from Synthetic Basis is Topology/Open Set Axiom 2 | Let $\BB$ be a synthetic basis on a set $X$.
Let $\ds \tau = \set {\bigcup \AA: \AA \subseteq \BB}$.
Then $\tau$ fulfils {{Open-set-axiom|2}}. | 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 $\tau$ fulfils {{Open-set-axiom|2}}. | 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/Open_Set_Axiom_2 | https://proofwiki.org/wiki/Union_from_Synthetic_Basis_is_Topology/Open_Set_Axiom_2/Proof_1 | [
"Union from Synthetic Basis is Topology"
] | [
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Set"
] | [
"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-23207 | Union from Synthetic Basis is Topology/Open Set Axiom 3 | Let $\BB$ be a synthetic basis on a set $X$.
Let $\ds \tau = \set {\bigcup \AA: \AA \subseteq \BB}$.
Then $\tau$ fulfils {{Open-set-axiom|3}}. | 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 $\tau$ fulfils {{Open-set-axiom|3}}. | 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/Open_Set_Axiom_3 | https://proofwiki.org/wiki/Union_from_Synthetic_Basis_is_Topology/Open_Set_Axiom_3/Proof_1 | [
"Union from Synthetic Basis is Topology"
] | [
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Set"
] | [
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Cover of Set",
"Equivalent Conditions for Cover by Collection of Subsets"
] |
proofwiki-23208 | Union from Synthetic Basis is Topology/Open Set Axiom 3 | Let $\BB$ be a synthetic basis on a set $X$.
Let $\ds \tau = \set {\bigcup \AA: \AA \subseteq \BB}$.
Then $\tau$ fulfils {{Open-set-axiom|3}}. | 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 $\tau$ fulfils {{Open-set-axiom|3}}. | 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/Open_Set_Axiom_3 | https://proofwiki.org/wiki/Union_from_Synthetic_Basis_is_Topology/Open_Set_Axiom_3/Proof_2 | [
"Union from Synthetic Basis is Topology"
] | [
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Set"
] | [
"Definition:Basis (Topology)/Synthetic Basis"
] |
proofwiki-23209 | Union from Synthetic Basis is Topology/Open Set Axiom 1 | Let $\BB$ be a synthetic basis on a set $X$.
Let $\ds \tau = \set {\bigcup \AA: \AA \subseteq \BB}$.
Then $\tau$ fulfils {{Open-set-axiom|1}}. | 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 $\tau$ fulfils {{Open-set-axiom|1}}. | 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/Open_Set_Axiom_1 | https://proofwiki.org/wiki/Union_from_Synthetic_Basis_is_Topology/Open_Set_Axiom_1/Proof_1 | [
"Union from Synthetic Basis is Topology"
] | [
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Set"
] | [
"Union is Smallest Superset/Family of Sets",
"Equivalence of Definitions of Topology Generated by Synthetic Basis",
"Set Union is Self-Distributive/General Result"
] |
proofwiki-23210 | Square Number ending in 10 Digits in Reverse Order | The next integer after $1 \, 111 \, 111 \, 111$ whose square ends in $0 \, 987 \, 654 \, 321$ is:
:$2 \, 380 \, 642 \, 361^2 = 5 \, 667 \, 458 \, 050 \, 987 \, 654 \, 321$ | {{ProofWanted|Is there a way to do this without an exhaustive search?}} | The next [[Definition:Integer|integer]] after $1 \, 111 \, 111 \, 111$ whose [[Definition:Square (Algebra)|square]] ends in $0 \, 987 \, 654 \, 321$ is:
:$2 \, 380 \, 642 \, 361^2 = 5 \, 667 \, 458 \, 050 \, 987 \, 654 \, 321$ | {{ProofWanted|Is there a way to do this without an exhaustive search?}} | Square Number ending in 10 Digits in Reverse Order | https://proofwiki.org/wiki/Square_Number_ending_in_10_Digits_in_Reverse_Order | https://proofwiki.org/wiki/Square_Number_ending_in_10_Digits_in_Reverse_Order | [
"Square Numbers"
] | [
"Definition:Integer",
"Definition:Square/Function"
] | [] |
proofwiki-23211 | Continued Fraction for Pi over N Times Real Cotangent Function | :$\dfrac \pi n \map \cot {\dfrac {\pi m} n} = \cfrac 1 {m + \cfrac {m^2} {n - 2 m + \cfrac {\paren {n - m}^2} {2 m + \cfrac {\paren {n + m}^2} {n - 2 m + \cfrac {\paren {2 n - m}^2} {2 m + \cfrac {\paren {2 n + m}^2} \ddots} } } } }$ | From {{Corollary|Mittag-Leffler Expansion for Cotangent Function|2}}, we have:
{{begin-eqn}}
{{eqn | l = \frac \pi n \map \cot {\frac {\pi m} n}
| r = \frac 1 m + \sum_{k \mathop = 1}^\infty \paren {\frac 1 {k n + m} - \frac 1 {k n - m} }
| c =
}}
{{eqn | r = \frac 1 m + \paren {\frac 1 {n + m} - \frac 1 {... | :$\dfrac \pi n \map \cot {\dfrac {\pi m} n} = \cfrac 1 {m + \cfrac {m^2} {n - 2 m + \cfrac {\paren {n - m}^2} {2 m + \cfrac {\paren {n + m}^2} {n - 2 m + \cfrac {\paren {2 n - m}^2} {2 m + \cfrac {\paren {2 n + m}^2} \ddots} } } } }$ | From {{Corollary|Mittag-Leffler Expansion for Cotangent Function|2}}, we have:
{{begin-eqn}}
{{eqn | l = \frac \pi n \map \cot {\frac {\pi m} n}
| r = \frac 1 m + \sum_{k \mathop = 1}^\infty \paren {\frac 1 {k n + m} - \frac 1 {k n - m} }
| c =
}}
{{eqn | r = \frac 1 m + \paren {\frac 1 {n + m} - \frac 1 ... | Continued Fraction for Pi over N Times Real Cotangent Function | https://proofwiki.org/wiki/Continued_Fraction_for_Pi_over_N_Times_Real_Cotangent_Function | https://proofwiki.org/wiki/Continued_Fraction_for_Pi_over_N_Times_Real_Cotangent_Function | [
"Cotangent Function",
"Continued Fractions",
"Euler's Continued Fraction Formula",
"Examples of Euler's Continued Fraction Formula",
"Mittag-Leffler Expansion for Cotangent Function",
"Mittag-Leffler Expansions"
] | [] | [] |
proofwiki-23212 | Cotangent of 22.5 Degrees | :$\cot 22.5 \degrees = \cot \dfrac \pi 8 = \sqrt 2 + 1$ | {{begin-eqn}}
{{eqn | l = \cot 22.5 \degrees
| r = \frac {\cos 22.5 \degrees} {\sin 22.5 \degrees}
| c = {{Defof|Cotangent/Complex Function|Cotangent}}
}}
{{eqn | r = \frac {\frac 1 2 \sqrt {2 + \sqrt 2} } {\frac 1 2 \sqrt {2 - \sqrt 2} }
| c = {{sin|22.5}}, {{cos|22.5}}
}}
{{eqn | r = \frac {\sqrt {2... | :$\cot 22.5 \degrees = \cot \dfrac \pi 8 = \sqrt 2 + 1$ | {{begin-eqn}}
{{eqn | l = \cot 22.5 \degrees
| r = \frac {\cos 22.5 \degrees} {\sin 22.5 \degrees}
| c = {{Defof|Cotangent/Complex Function|Cotangent}}
}}
{{eqn | r = \frac {\frac 1 2 \sqrt {2 + \sqrt 2} } {\frac 1 2 \sqrt {2 - \sqrt 2} }
| c = {{sin|22.5}}, {{cos|22.5}}
}}
{{eqn | r = \frac {\sqrt {2... | Cotangent of 22.5 Degrees/Proof 1 | https://proofwiki.org/wiki/Cotangent_of_22.5_Degrees | https://proofwiki.org/wiki/Cotangent_of_22.5_Degrees/Proof_1 | [
"Cotangent of 22.5 Degrees",
"Cotangent Function"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Two Squares",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-23213 | Closure and Complement Operations generate Monoid | Let $T = \struct {S, \tau}$ be a topological space.
Let $\AA$ be the set of subspaces of $T$.
Let $\Phi: \AA \to \AA$ be the set of sequences of compositions of mappings consisting of:
:complementation relative to $S$
:topological closure
:the identity mapping.
Let $\otimes: \Phi \times \Phi \to \Phi$ denote the binary... | The fact that there are at most $14$ elements is demonstrated in the proof of maximum of Kuratowski's Closure-Complement Problem.
From that same page, we use the notation:
:let $a$ be used to denote the operation of taking the complement of $A$ relative to $S$: $\map a A = S \setminus A$
:let $b$ be used to denote the ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\AA$ be the [[Definition:Set of Sets|set]] of [[Definition:Topological Subspace|subspaces]] of $T$.
Let $\Phi: \AA \to \AA$ be the [[Definition:Set|set]] of [[Definition:Sequence|sequences]] of [[Definition:Composition of Mappi... | The fact that there are at most $14$ [[Definition:Element|elements]] is demonstrated in the [[Kuratowski's Closure-Complement Problem/Proof of Maximum|proof of maximum]] of [[Kuratowski's Closure-Complement Problem]].
From that same page, we use the notation:
:let $a$ be used to denote the [[Definition:Unary Operation... | Closure and Complement Operations generate Monoid | https://proofwiki.org/wiki/Closure_and_Complement_Operations_generate_Monoid | https://proofwiki.org/wiki/Closure_and_Complement_Operations_generate_Monoid | [
"Closure and Complement Operations generate Monoid",
"Kuratowski's Closure-Complement Problem",
"Set Closures",
"Relative Complement",
"Examples of Monoids"
] | [
"Definition:Topological Space",
"Definition:Set of Sets",
"Definition:Topological Subspace",
"Definition:Set",
"Definition:Sequence",
"Definition:Composition of Mappings",
"Definition:Mapping",
"Definition:Relative Complement",
"Definition:Closure (Topology)",
"Definition:Identity Mapping",
"Def... | [
"Definition:Element",
"Kuratowski's Closure-Complement Problem/Proof of Maximum",
"Kuratowski's Closure-Complement Problem",
"Definition:Operation/Unary Operation",
"Definition:Relative Complement",
"Definition:Operation/Unary Operation",
"Definition:Closure (Topology)",
"Definition:Identity Mapping",... |
proofwiki-23214 | Regular Open Set is Open | Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$ be regular open in $T$.
Then $H$ is an open set of $T$. | {{Recall|Regular Open Set|regular open set}}
{{:Definition:Regular Open Set}}
{{Recall|Interior (Topology)|set interior}}
{{:Definition:Interior (Topology)/Definition 2}}
Hence:
:as $H$ equals its interior
and:
:the interior of $H$ is an open set of $T$ by definition
the result follows.
{{qed}}
Category:Regular Open Se... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq S$ be [[Definition:Regular Open Set|regular open]] in $T$.
Then $H$ is an [[Definition:Open Set (Topology)|open set]] of $T$. | {{Recall|Regular Open Set|regular open set}}
{{:Definition:Regular Open Set}}
{{Recall|Interior (Topology)|set interior}}
{{:Definition:Interior (Topology)/Definition 2}}
Hence:
:as $H$ equals its [[Definition:Interior (Topology)|interior]]
and:
:the [[Definition:Interior (Topology)|interior]] of $H$ is an [[Definiti... | Regular Open Set is Open | https://proofwiki.org/wiki/Regular_Open_Set_is_Open | https://proofwiki.org/wiki/Regular_Open_Set_is_Open | [
"Regular Open Sets",
"Open Sets"
] | [
"Definition:Topological Space",
"Definition:Regular Open Set",
"Definition:Open Set/Topology"
] | [
"Definition:Interior (Topology)",
"Definition:Interior (Topology)",
"Definition:Open Set/Topology",
"Category:Regular Open Sets",
"Category:Open Sets"
] |
proofwiki-23215 | Regular Closed Set is Closed | Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$ be regular closed in $T$.
Then $H$ is an closed set of $T$. | {{Recall|Regular Closed Set|regular closed set}}
{{:Definition:Regular Closed Set}}
{{Recall|Closure (Topology)|set closure}}
{{:Definition:Closure (Topology)/Definition 3}}
Hence:
:as $H$ equals the closure of its interior
the result follows from Topological Closure is Closed.
{{qed}}
Category:Regular Closed Sets
Cate... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq S$ be [[Definition:Regular Closed Set|regular closed]] in $T$.
Then $H$ is an [[Definition:Closed Set (Topology)|closed set]] of $T$. | {{Recall|Regular Closed Set|regular closed set}}
{{:Definition:Regular Closed Set}}
{{Recall|Closure (Topology)|set closure}}
{{:Definition:Closure (Topology)/Definition 3}}
Hence:
:as $H$ equals the [[Definition:Closure (Topology)|closure]] of its [[Definition:Interior (Topology)|interior]]
the result follows from ... | Regular Closed Set is Closed | https://proofwiki.org/wiki/Regular_Closed_Set_is_Closed | https://proofwiki.org/wiki/Regular_Closed_Set_is_Closed | [
"Regular Closed Sets",
"Closed Sets"
] | [
"Definition:Topological Space",
"Definition:Regular Closed Set",
"Definition:Closed Set/Topology"
] | [
"Definition:Closure (Topology)",
"Definition:Interior (Topology)",
"Topological Closure is Closed",
"Category:Regular Closed Sets",
"Category:Closed Sets"
] |
proofwiki-23216 | Continuity Defined by Closure/Necessary Condition | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $f: T_1 \to T_2$ be a mapping.
Let $f$ be continuous.
Then:
:$\forall H \subseteq S_1: f \sqbrk {H^-} \subseteq \paren {f \sqbrk H}^-$
where $H^-$ denotes the closure of $H$ in $T_1$.
That is, the image of the closure is a su... | Let $f$ be continuous.
Let $y \in f \sqbrk {\map \cl H}$.
Then:
:$\exists x \in \map \cl H: y = \map f x$
Let $U$ be an open set of $T_2$ such that $y \in U$.
Then by definition of continuous mapping:
:$f^{-1} \sqbrk U$ is an open set of $T_1$ such that:
::$x \in f^{-1} \sqbrk U$
Hence:
:$f^{-1} \sqbrk U \cap H \ne \O$... | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $f: T_1 \to T_2$ be a [[Definition:Mapping|mapping]].
Let $f$ be [[Definition:Everywhere Continuous Mapping (Topology)|continuous]].
Then:
:$\forall H \subseteq S_1: f \sqbrk {H^-} \subset... | Let $f$ be [[Definition:Everywhere Continuous Mapping (Topology)|continuous]].
Let $y \in f \sqbrk {\map \cl H}$.
Then:
:$\exists x \in \map \cl H: y = \map f x$
Let $U$ be an [[Definition:Open Set (Topology)|open set]] of $T_2$ such that $y \in U$.
Then by definition of [[Definition:Everywhere Continuous Mapping ... | Continuity Defined by Closure/Necessary Condition/Proof 1 | https://proofwiki.org/wiki/Continuity_Defined_by_Closure/Necessary_Condition | https://proofwiki.org/wiki/Continuity_Defined_by_Closure/Necessary_Condition/Proof_1 | [
"Continuity Defined by Closure"
] | [
"Definition:Topological Space",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Closure (Topology)",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Closure (Topology)",
"Definition:Subset",
"Definition:Closure (Topology)",
"Definition:Image (Set ... | [
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Open Set/Topology",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Open Set/Topology"
] |
proofwiki-23217 | Regular Space is not necessarily Normal | A topological space which is a regular space is not necessarily also a normal space. | Let $T$ be a Niemytzki plane.
From Niemytzki Plane is Completely Regular, $T$ is a completely regular space.
Hence by Completely Regular Space is Regular, $T$ is a regular space.
But by Niemytzki Plane is not Normal, $T$ is not a normal space.
Hence the result.
{{qed}} | A [[Definition:Topological Space|topological space]] which is a [[Definition:Regular Space|regular space]] is not necessarily also a [[Definition:Normal Space|normal space]]. | Let $T$ be a [[Definition:Niemytzki Plane|Niemytzki plane]].
From [[Niemytzki Plane is Completely Regular]], $T$ is a [[Definition:Completely Regular Space|completely regular space]].
Hence by [[Completely Regular Space is Regular]], $T$ is a [[Definition:Regular Space|regular space]].
But by [[Niemytzki Plane is n... | Regular Space is not necessarily Normal | https://proofwiki.org/wiki/Regular_Space_is_not_necessarily_Normal | https://proofwiki.org/wiki/Regular_Space_is_not_necessarily_Normal | [
"Regular Spaces",
"Normal Spaces"
] | [
"Definition:Topological Space",
"Definition:Regular Space",
"Definition:Normal Space"
] | [
"Definition:Niemytzki Plane",
"Niemytzki Plane is Completely Regular",
"Definition:Completely Regular Space",
"Completely Regular Space is Regular",
"Definition:Regular Space",
"Niemytzki Plane is not Normal",
"Definition:Normal Space"
] |
proofwiki-23218 | T2 Space is not necessarily Regular | A topological space which is a $T_2$ (Hausdorff) space is not necessarily also a regular space. | Let $T$ be an irrational slope topological space.
From Irrational Slope Space is $T_2$, $T$ is a $T_2$ (Hausdorff) space.
But from Irrational Slope Space is not Regular, $T$ is not a regular space.
Hence the result.
{{qed}} | A [[Definition:Topological Space|topological space]] which is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]] is not necessarily also a [[Definition:Regular Space|regular space]]. | Let $T$ be an [[Definition:Irrational Slope Topology|irrational slope topological space]].
From [[Irrational Slope Space is T2|Irrational Slope Space is $T_2$]], $T$ is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
But from [[Irrational Slope Space is not Regular]], $T$ is not a [[Definition:Regular Space|regula... | T2 Space is not necessarily Regular | https://proofwiki.org/wiki/T2_Space_is_not_necessarily_Regular | https://proofwiki.org/wiki/T2_Space_is_not_necessarily_Regular | [
"Hausdorff Spaces",
"Regular Spaces"
] | [
"Definition:Topological Space",
"Definition:T2 Space",
"Definition:Regular Space"
] | [
"Definition:Irrational Slope Topology",
"Irrational Slope Space is T2",
"Definition:T2 Space",
"Irrational Slope Space is not Regular",
"Definition:Regular Space"
] |
proofwiki-23219 | T1 Space is not necessarily T2 | A topological space which is a $T_1$ space is not necessarily also a $T_2$ (Hausdorff) space. | Let $T = \struct {S, \tau}$ be the finite complement topology on an infinite set $S$.
From Finite Complement Space is $T_1$, $T$ is a $T_1$ space.
But from Finite Complement Space is not $T_2$, $T$ is not a $T_2$ (Hausdorff) space.
Hence the result.
{{qed}} | A [[Definition:Topological Space|topological space]] which is a [[Definition:T1 Space|$T_1$ space]] is not necessarily also a [[Definition:T2 Space|$T_2$ (Hausdorff) space]]. | Let $T = \struct {S, \tau}$ be the [[Definition:Finite Complement Topology|finite complement topology]] on an [[Definition:Infinite Set|infinite]] set $S$.
From [[Finite Complement Space is T1|Finite Complement Space is $T_1$]], $T$ is a [[Definition:T1 Space|$T_1$ space]].
But from [[Finite Complement Space is not ... | T1 Space is not necessarily T2 | https://proofwiki.org/wiki/T1_Space_is_not_necessarily_T2 | https://proofwiki.org/wiki/T1_Space_is_not_necessarily_T2 | [
"T1 Spaces",
"Hausdorff Spaces"
] | [
"Definition:Topological Space",
"Definition:T1 Space",
"Definition:T2 Space"
] | [
"Definition:Finite Complement Topology",
"Definition:Infinite Set",
"Finite Complement Space is T1",
"Definition:T1 Space",
"Finite Complement Space is not T2",
"Definition:T2 Space"
] |
proofwiki-23220 | T4 and T0 Space is not necessarily T3 | A topological space which is both a $T_4$ space and a $T_0$ space is not necessarily also a $T_3$ space. | Let $T$ be a Hjalmar Ekdal space.
From Hjalmar Ekdal Space is $T_4$, $T$ is a $T_4$ space.
From Hjalmar Ekdal Space is $T_0$, $T$ is a $T_0$ space.
But from Hjalmar Ekdal Space is not $T_3$, $T$ is not a $T_3$ space.
Hence the result.
{{qed}} | A [[Definition:Topological Space|topological space]] which is both a [[Definition:T4 Space|$T_4$ space]] and a [[Definition:T0 Space|$T_0$ space]] is not necessarily also a [[Definition:T3 Space|$T_3$ space]]. | Let $T$ be a [[Definition:Hjalmar Ekdal Topology|Hjalmar Ekdal space]].
From [[Hjalmar Ekdal Space is T4|Hjalmar Ekdal Space is $T_4$]], $T$ is a [[Definition:T4 Space|$T_4$ space]].
From [[Hjalmar Ekdal Space is T0|Hjalmar Ekdal Space is $T_0$]], $T$ is a [[Definition:T0 Space|$T_0$ space]].
But from [[Hjalmar Ekd... | T4 and T0 Space is not necessarily T3 | https://proofwiki.org/wiki/T4_and_T0_Space_is_not_necessarily_T3 | https://proofwiki.org/wiki/T4_and_T0_Space_is_not_necessarily_T3 | [
"T4 Spaces",
"T0 Spaces",
"T3 Spaces"
] | [
"Definition:Topological Space",
"Definition:T4 Space",
"Definition:T0 Space",
"Definition:T3 Space"
] | [
"Definition:Hjalmar Ekdal Topology",
"Hjalmar Ekdal Space is T4",
"Definition:T4 Space",
"Hjalmar Ekdal Space is T0",
"Definition:T0 Space",
"Hjalmar Ekdal Space is not T3",
"Definition:T3 Space"
] |
proofwiki-23221 | T5 and T0 Space is not necessarily T3 | A topological space which is both a $T_5$ space and a $T_0$ space is not necessarily also a $T_3$ space. | Let $T$ be a Hjalmar Ekdal space.
From Hjalmar Ekdal Space is $T_5$, $T$ is a $T_5$ space.
From Hjalmar Ekdal Space is $T_0$, $T$ is a $T_0$ space.
But from Hjalmar Ekdal Space is not $T_3$, $T$ is not a $T_3$ space.
Hence the result.
{{qed}} | A [[Definition:Topological Space|topological space]] which is both a [[Definition:T5 Space|$T_5$ space]] and a [[Definition:T0 Space|$T_0$ space]] is not necessarily also a [[Definition:T3 Space|$T_3$ space]]. | Let $T$ be a [[Definition:Hjalmar Ekdal Topology|Hjalmar Ekdal space]].
From [[Hjalmar Ekdal Space is T5|Hjalmar Ekdal Space is $T_5$]], $T$ is a [[Definition:T5 Space|$T_5$ space]].
From [[Hjalmar Ekdal Space is T0|Hjalmar Ekdal Space is $T_0$]], $T$ is a [[Definition:T0 Space|$T_0$ space]].
But from [[Hjalmar Ekd... | T5 and T0 Space is not necessarily T3 | https://proofwiki.org/wiki/T5_and_T0_Space_is_not_necessarily_T3 | https://proofwiki.org/wiki/T5_and_T0_Space_is_not_necessarily_T3 | [
"T5 Spaces",
"T0 Spaces",
"T3 Spaces"
] | [
"Definition:Topological Space",
"Definition:T5 Space",
"Definition:T0 Space",
"Definition:T3 Space"
] | [
"Definition:Hjalmar Ekdal Topology",
"Hjalmar Ekdal Space is T5",
"Definition:T5 Space",
"Hjalmar Ekdal Space is T0",
"Definition:T0 Space",
"Hjalmar Ekdal Space is not T3",
"Definition:T3 Space"
] |
proofwiki-23222 | Half-Disc Space is not Regular | Let $T$ be a half-disc space.
Then $T$ is not a regular space. | From Half-Disc Space is not Semiregular, $T$ is not a semiregular space.
From Regular Space is Semiregular, it follows that $T$ cannot therefore be a regular space.
{{qed}}
Category:Half-Disc Topology
Category:Examples of Regular Spaces
a4q6tpdx79yttpnj6m82mvxjzjzjcs2 | Let $T$ be a [[Definition:Half-Disc Space|half-disc space]].
Then $T$ is not a [[Definition:Regular Space|regular space]]. | From [[Half-Disc Space is not Semiregular]], $T$ is not a [[Definition:Semiregular Space|semiregular space]].
From [[Regular Space is Semiregular]], it follows that $T$ cannot therefore be a [[Definition:Regular Space|regular space]].
{{qed}}
[[Category:Half-Disc Topology]]
[[Category:Examples of Regular Spaces]]
a4q... | Half-Disc Space is not Regular | https://proofwiki.org/wiki/Half-Disc_Space_is_not_Regular | https://proofwiki.org/wiki/Half-Disc_Space_is_not_Regular | [
"Half-Disc Topology",
"Examples of Regular Spaces"
] | [
"Definition:Half-Disc Topology",
"Definition:Regular Space"
] | [
"Half-Disc Space is not Semiregular",
"Definition:Semiregular Space",
"Regular Space is Semiregular",
"Definition:Regular Space",
"Category:Half-Disc Topology",
"Category:Examples of Regular Spaces"
] |
proofwiki-23223 | Overlapping Interval Space fulfils no Separation Axioms but T0 | Let $T$ be the overlapping interval space.
Then $T$ fulfils none of the Tychonoff separation axioms but the $T_0$ axiom. | From Overlapping Interval Space is $T_0$ we note that indeed $T$ is a $T_0$ space.
From Overlapping Interval Space is not $T_1$, $T$ is not a $T_1$ space.
From Overlapping Interval Space is not $T_4$, $T$ is not a $T_4$ space.
The result follows from Sequence of Implications of Separation Axioms and the Rule of Transpo... | Let $T$ be the [[Definition:Overlapping Interval Space|overlapping interval space]].
Then $T$ fulfils none of the [[Definition:Tychonoff Separation Axioms|Tychonoff separation axioms]] but the [[Definition:T0 Space|$T_0$ axiom]]. | From [[Overlapping Interval Space is T0|Overlapping Interval Space is $T_0$]] we note that indeed $T$ is a [[Definition:T0 Space|$T_0$ space]].
From [[Overlapping Interval Space is not T1|Overlapping Interval Space is not $T_1$]], $T$ is not a [[Definition:T1 Space|$T_1$ space]].
From [[Overlapping Interval Space is ... | Overlapping Interval Space fulfils no Separation Axioms but T0 | https://proofwiki.org/wiki/Overlapping_Interval_Space_fulfils_no_Separation_Axioms_but_T0 | https://proofwiki.org/wiki/Overlapping_Interval_Space_fulfils_no_Separation_Axioms_but_T0 | [
"Overlapping Interval Topology",
"Separation Axioms"
] | [
"Definition:Overlapping Interval Topology",
"Definition:Tychonoff Separation Axioms",
"Definition:T0 Space"
] | [
"Overlapping Interval Space is T0",
"Definition:T0 Space",
"Overlapping Interval Space is not T1",
"Definition:T1 Space",
"Overlapping Interval Space is not T4",
"Definition:T4 Space",
"Sequence of Implications of Separation Axioms",
"Rule of Transposition",
"Definition:T4 Space",
"T5 Space is T4"... |
proofwiki-23224 | Overlapping Interval Space is T0 | Let $T$ be the overlapping interval space.
Then $T$ is a $T_0$ space. | Let $a, b \in \closedint {-1} 1$ such that $a < b$.
Let $c \in \closedint {-1} 1$ such that $a < c < b$ but $c \ne 0$.
;Proof by Cases
$(1): \quad$ Let $c < 0$.
By Open Sets of Overlapping Interval Space, $U := \hointl c 1$ is an open set of $T$.
Thus:
:$a \notin U$
but
:$b \in U$
and we see that $U$ fulfils the condit... | Let $T$ be the [[Definition:Overlapping Interval Space|overlapping interval space]].
Then $T$ is a [[Definition:T0 Space|$T_0$ space]]. | Let $a, b \in \closedint {-1} 1$ such that $a < b$.
Let $c \in \closedint {-1} 1$ such that $a < c < b$ but $c \ne 0$.
;[[Proof by Cases]]
$(1): \quad$ Let $c < 0$.
By [[Open Sets of Overlapping Interval Space]], $U := \hointl c 1$ is an [[Definition:Open Set (Topology)|open set]] of $T$.
Thus:
:$a \notin U$
but
... | Overlapping Interval Space is T0 | https://proofwiki.org/wiki/Overlapping_Interval_Space_is_T0 | https://proofwiki.org/wiki/Overlapping_Interval_Space_is_T0 | [
"Overlapping Interval Topology",
"Examples of T0 Spaces"
] | [
"Definition:Overlapping Interval Topology",
"Definition:T0 Space"
] | [
"Proof by Cases",
"Open Sets of Overlapping Interval Space",
"Definition:Open Set/Topology",
"Definition:Element",
"Definition:T0 Space",
"Open Sets of Overlapping Interval Space",
"Definition:Open Set/Topology",
"Definition:Element",
"Definition:T0 Space",
"Definition:Arbitrary"
] |
proofwiki-23225 | Overlapping Interval Space is not T1 | Let $T$ be the overlapping interval space.
Then $T$ is not a $T_1$ space. | {{Recall|T1 Space|index = 1|$T_1$ space}}
{{:Definition:T1 Space/Definition 1}}
Let $x \in S$ be arbitrary such that $x \ne 0$.
Consider the element $0 \in S$.
Let $U \in \tau$ such that $U \ne \O$.
From Open Sets of Overlapping Interval Space, $U$ is of the form of one of:
{{begin-axiom}}
{{axiom | n = 1
| m =... | Let $T$ be the [[Definition:Overlapping Interval Space|overlapping interval space]].
Then $T$ is not a [[Definition:T1 Space|$T_1$ space]]. | {{Recall|T1 Space|index = 1|$T_1$ space}}
{{:Definition:T1 Space/Definition 1}}
Let $x \in S$ be [[Definition:Arbitrary|arbitrary]] such that $x \ne 0$.
Consider the [[Definition:Element|element]] $0 \in S$.
Let $U \in \tau$ such that $U \ne \O$.
From [[Open Sets of Overlapping Interval Space]], $U$ is of the form ... | Overlapping Interval Space is not T1 | https://proofwiki.org/wiki/Overlapping_Interval_Space_is_not_T1 | https://proofwiki.org/wiki/Overlapping_Interval_Space_is_not_T1 | [
"Overlapping Interval Topology",
"Examples of T1 Spaces"
] | [
"Definition:Overlapping Interval Topology",
"Definition:T1 Space"
] | [
"Definition:Arbitrary",
"Definition:Element",
"Open Sets of Overlapping Interval Space"
] |
proofwiki-23226 | Overlapping Interval Space is not T4 | Let $T$ be the overlapping interval space.
Then $T$ is not a $T_4$ space. | {{Recall|T4 Space|index = 1|$T_4$ space}}
{{:Definition:T4 Space/Definition 1}}
Consider the sets $\set {-1}$ and $\set 1$, both of which are subsets of $S$.
We have that:
{{begin-eqn}}
{{eqn | l = \relcomp S {\set {-1} }
| r = \hointl {-1} 1
}}
{{eqn | l = \relcomp S {\set 1}
| r = \hointr {-1} 1
}}
{{end-... | Let $T$ be the [[Definition:Overlapping Interval Space|overlapping interval space]].
Then $T$ is not a [[Definition:T4 Space|$T_4$ space]]. | {{Recall|T4 Space|index = 1|$T_4$ space}}
{{:Definition:T4 Space/Definition 1}}
Consider the [[Definition:Set|sets]] $\set {-1}$ and $\set 1$, both of which are [[Definition:Subset|subsets]] of $S$.
We have that:
{{begin-eqn}}
{{eqn | l = \relcomp S {\set {-1} }
| r = \hointl {-1} 1
}}
{{eqn | l = \relcomp S {... | Overlapping Interval Space is not T4 | https://proofwiki.org/wiki/Overlapping_Interval_Space_is_not_T4 | https://proofwiki.org/wiki/Overlapping_Interval_Space_is_not_T4 | [
"Overlapping Interval Topology",
"Examples of T4 Spaces"
] | [
"Definition:Overlapping Interval Topology",
"Definition:T4 Space"
] | [
"Definition:Set",
"Definition:Subset",
"Open Sets of Overlapping Interval Space",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Open Sets of Overlapping Interval Space",
"Definition:Open Set/Topology",
"Definition:Subset",
"Definition:Open Set/Topology",
"Definition:Subset",
... |
proofwiki-23227 | Hjalmar Ekdal Space is not Regular | Let $T = \struct {S, \tau}$ be a Hjalmar Ekdal space.
Then $T$ is ''not'' a regular space. | {{Recall|Regular Space|regular space|index = 2}}
{{:Definition:Regular Space/Definition 2}}
Let $T = \struct {S, \tau}$ be a Hjalmar Ekdal space.
But from Hjalmar Ekdal Space is not $T_1$, $T$ is not a $T_1$ space.
Hence the result.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Hjalmar Ekdal Space|Hjalmar Ekdal space]].
Then $T$ is ''not'' a [[Definition:Regular Space|regular space]]. | {{Recall|Regular Space|regular space|index = 2}}
{{:Definition:Regular Space/Definition 2}}
Let $T = \struct {S, \tau}$ be a [[Definition:Hjalmar Ekdal Space|Hjalmar Ekdal space]].
But from [[Hjalmar Ekdal Space is not T1|Hjalmar Ekdal Space is not $T_1$]], $T$ is not a [[Definition:T1 Space|$T_1$ space]].
Hence the... | Hjalmar Ekdal Space is not Regular | https://proofwiki.org/wiki/Hjalmar_Ekdal_Space_is_not_Regular | https://proofwiki.org/wiki/Hjalmar_Ekdal_Space_is_not_Regular | [
"Hjalmar Ekdal Topology",
"Examples of Regular Spaces"
] | [
"Definition:Hjalmar Ekdal Topology",
"Definition:Regular Space"
] | [
"Definition:Hjalmar Ekdal Topology",
"Hjalmar Ekdal Space is not T1",
"Definition:T1 Space"
] |
proofwiki-23228 | Hjalmar Ekdal Space is not T3.5 | Let $T$ be a Hjalmar Ekdal space.
Then $T$ is ''not'' a $T_{3 \frac 1 2}$ space. | We have that a $T_{3 \frac 1 2}$ Space is $T_3$.
From Hjalmar Ekdal Space is not $T_3$, $T$ is not a $T_3$ space.
The result follows by applying the Rule of Transposition.
{{qed}} | Let $T$ be a [[Definition:Hjalmar Ekdal Space|Hjalmar Ekdal space]].
Then $T$ is ''not'' a [[Definition:T3.5 Space|$T_{3 \frac 1 2}$ space]]. | We have that a [[T3.5 Space is T3|$T_{3 \frac 1 2}$ Space is $T_3$]].
From [[Hjalmar Ekdal Space is not T3|Hjalmar Ekdal Space is not $T_3$]], $T$ is not a [[Definition:T3 Space|$T_3$ space]].
The result follows by applying the [[Rule of Transposition]].
{{qed}} | Hjalmar Ekdal Space is not T3.5 | https://proofwiki.org/wiki/Hjalmar_Ekdal_Space_is_not_T3.5 | https://proofwiki.org/wiki/Hjalmar_Ekdal_Space_is_not_T3.5 | [
"Hjalmar Ekdal Topology",
"Examples of T3.5 Spaces"
] | [
"Definition:Hjalmar Ekdal Topology",
"Definition:T3.5 Space"
] | [
"T3.5 Space is T3",
"Hjalmar Ekdal Space is not T3",
"Definition:T3 Space",
"Rule of Transposition"
] |
proofwiki-23229 | Hjalmar Ekdal Space is not T3 | Let $T = \struct {S, \tau}$ be a Hjalmar Ekdal space.
Then $T$ is ''not'' a $T_3$ space. | {{AimForCont}} $T = \struct {S, \tau}$ were a $T_3$ space.
From Hjalmar Ekdal Space is $T_0$, $T$ is a $T_0$ space.
Then, by definition, $T$ would then be a regular space.
But from Hjalmar Ekdal Space is not Regular, $T$ is not a regular space.
The result follows by Proof by Contradiction.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Hjalmar Ekdal Space|Hjalmar Ekdal space]].
Then $T$ is ''not'' a [[Definition:T3 Space|$T_3$ space]]. | {{AimForCont}} $T = \struct {S, \tau}$ were a [[Definition:T3 Space|$T_3$ space]].
From [[Hjalmar Ekdal Space is T0|Hjalmar Ekdal Space is $T_0$]], $T$ is a [[Definition:T0 Space|$T_0$ space]].
Then, by definition, $T$ would then be a [[Definition:Regular Space|regular space]].
But from [[Hjalmar Ekdal Space is not... | Hjalmar Ekdal Space is not T3 | https://proofwiki.org/wiki/Hjalmar_Ekdal_Space_is_not_T3 | https://proofwiki.org/wiki/Hjalmar_Ekdal_Space_is_not_T3 | [
"Hjalmar Ekdal Topology",
"Examples of T3 Spaces"
] | [
"Definition:Hjalmar Ekdal Topology",
"Definition:T3 Space"
] | [
"Definition:T3 Space",
"Hjalmar Ekdal Space is T0",
"Definition:T0 Space",
"Definition:Regular Space",
"Hjalmar Ekdal Space is not Regular",
"Definition:Regular Space",
"Proof by Contradiction"
] |
proofwiki-23230 | Hjalmar Ekdal Space is not T1 | Let $T = \struct {S, \tau}$ be a Hjalmar Ekdal space.
Then $T$ is ''not'' a $T_1$ space. | ;Proof by Counterexample
Consider the integers $1$ and $2$, both of which are elements of $S$.
Let $U \in S$ such that $1 \in U$.
Then, by definition of $T$, $2 \in U$.
Hence there exists no $U \in S$ such that $1 \in U$ but $2 \notin U$.
Hence the result by definition of $T_1$ space.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Hjalmar Ekdal Space|Hjalmar Ekdal space]].
Then $T$ is ''not'' a [[Definition:T1 Space|$T_1$ space]]. | ;[[Proof by Counterexample]]
Consider the [[Definition:Integer|integers]] $1$ and $2$, both of which are [[Definition:Element|elements]] of $S$.
Let $U \in S$ such that $1 \in U$.
Then, by definition of $T$, $2 \in U$.
Hence there exists no $U \in S$ such that $1 \in U$ but $2 \notin U$.
Hence the result by defini... | Hjalmar Ekdal Space is not T1 | https://proofwiki.org/wiki/Hjalmar_Ekdal_Space_is_not_T1 | https://proofwiki.org/wiki/Hjalmar_Ekdal_Space_is_not_T1 | [
"Hjalmar Ekdal Topology",
"Examples of T1 Spaces"
] | [
"Definition:Hjalmar Ekdal Topology",
"Definition:T1 Space"
] | [
"Proof by Counterexample",
"Definition:Integer",
"Definition:Element",
"Definition:T1 Space"
] |
proofwiki-23231 | Normal Space is T3.5 | Let $\struct {S, \tau}$ be a normal space.
Then $\struct {S, \tau}$ is also a $T_{3 \frac 1 2}$ space. | {{Recall|T3.5 Space|$T_{3 \frac 1 2}$ Space}}
{{:Definition:T3.5 Space}}
{{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
Let $F$ be an arbitrary closed set of $T$.
Let $y \in S$ such that $y \notin F$.
{{Recall|T1 Space|$T_1$ Space|index = 3}}
{{:Definition:T1 Space/Definition 3... | Let $\struct {S, \tau}$ be a [[Definition:Normal Space|normal space]].
Then $\struct {S, \tau}$ is also a [[Definition:T3.5 Space|$T_{3 \frac 1 2}$ space]]. | {{Recall|T3.5 Space|$T_{3 \frac 1 2}$ Space}}
{{:Definition:T3.5 Space}}
{{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
Let $F$ be an arbitrary [[Definition:Closed Set (Topology)|closed set]] of $T$.
Let $y \in S$ such that $y \notin F$.
{{Recall|T1 Space|$T_1$ Space|index... | Normal Space is T3.5 | https://proofwiki.org/wiki/Normal_Space_is_T3.5 | https://proofwiki.org/wiki/Normal_Space_is_T3.5 | [
"Normal Spaces",
"T3.5 Spaces"
] | [
"Definition:Normal Space",
"Definition:T3.5 Space"
] | [
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:T4 Space",
"Definition:Disjoint Sets",
"Definition:Closed Set/Topology",
"Urysohn's Lemma",
"Definition:Urysohn Function",
"Definition:T3.5 Space"
] |
proofwiki-23232 | Exponential of Category of Sets | Let $\mathbf{Set}$ be the category of sets.
Let $B$ and $C$ be sets.
Let $C^B$ be the set of all mappings from $B$ to $C$.
Define a mapping $\epsilon : C^B \times B \to C$ by:
:$\map \epsilon {f,b} = \map f b$
where $C^B \times B$ denotes the Cartesian product of $C^B$ and $B$.
Then $\struct {C^B,\epsilon}$ is an expon... | {{ProofWanted}}
Category:Category of Sets
7zrpmqs5owokmkwi66pxmet2jgz6rut | Let $\mathbf{Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $B$ and $C$ be [[Definition:Set|sets]].
Let $C^B$ be the [[Definition:Set|set]] of all [[Definition:Mapping|mappings]] from $B$ to $C$.
Define a [[Definition:Mapping|mapping]] $\epsilon : C^B \times B \to C$ by:
:$\map \epsilon {f,b} = \... | {{ProofWanted}}
[[Category:Category of Sets]]
7zrpmqs5owokmkwi66pxmet2jgz6rut | Exponential of Category of Sets | https://proofwiki.org/wiki/Exponential_of_Category_of_Sets | https://proofwiki.org/wiki/Exponential_of_Category_of_Sets | [
"Category of Sets"
] | [
"Definition:Category of Sets",
"Definition:Set",
"Definition:Set",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Cartesian Product",
"Definition:Exponential (Category Theory)",
"Definition:Exponential (Category Theory)/Transpose"
] | [
"Category:Category of Sets"
] |
proofwiki-23233 | Tangent of 36 Degrees | :$\tan 36 \degrees = \tan \dfrac \pi 5 = 5^{\frac 1 4} \phi^{-\frac 3 2}$
where $\tan$ denotes the tangent function. | {{begin-eqn}}
{{eqn | l = \tan 36 \degrees
| r = \frac {\sin 36 \degrees} {\cos 36 \degrees}
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {2^{-1} 5^{\frac 1 4} \phi^{-\frac 1 2} } {2^{-1} \phi}
| c = {{sin|36}} and {{cos|36}}
}}
{{eqn | r = 5^{\frac 1 4} \phi^{-\frac 3 2}
| c =
... | :$\tan 36 \degrees = \tan \dfrac \pi 5 = 5^{\frac 1 4} \phi^{-\frac 3 2}$
where $\tan$ denotes the [[Definition:Tangent Function|tangent function]]. | {{begin-eqn}}
{{eqn | l = \tan 36 \degrees
| r = \frac {\sin 36 \degrees} {\cos 36 \degrees}
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {2^{-1} 5^{\frac 1 4} \phi^{-\frac 1 2} } {2^{-1} \phi}
| c = {{sin|36}} and {{cos|36}}
}}
{{eqn | r = 5^{\frac 1 4} \phi^{-\frac 3 2}
| c... | Tangent of 36 Degrees | https://proofwiki.org/wiki/Tangent_of_36_Degrees | https://proofwiki.org/wiki/Tangent_of_36_Degrees | [
"Golden Mean",
"Tangent Function"
] | [
"Definition:Tangent Function"
] | [
"Tangent is Sine divided by Cosine",
"Category:Golden Mean",
"Category:Tangent Function"
] |
proofwiki-23234 | Tangent of 72 Degrees | :$\tan 72 \degrees = \tan \dfrac {2 \pi} 5 = 5^{\frac 1 4} \phi^{\frac 3 2}$
where $\tan$ denotes the tangent function. | {{begin-eqn}}
{{eqn | l = \tan 72 \degrees
| r = \frac {\sin 72 \degrees} {\cos 72 \degrees}
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {2^{-1} 5^{\frac 1 4} \phi^{\frac 1 2} } {2^{-1} \phi^{-1} }
| c = {{sin|72}} and {{cos|72}}
}}
{{eqn | r = 5^{\frac 1 4} \phi^{\frac 3 2}
| c... | :$\tan 72 \degrees = \tan \dfrac {2 \pi} 5 = 5^{\frac 1 4} \phi^{\frac 3 2}$
where $\tan$ denotes the [[Definition:Tangent Function|tangent function]]. | {{begin-eqn}}
{{eqn | l = \tan 72 \degrees
| r = \frac {\sin 72 \degrees} {\cos 72 \degrees}
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {2^{-1} 5^{\frac 1 4} \phi^{\frac 1 2} } {2^{-1} \phi^{-1} }
| c = {{sin|72}} and {{cos|72}}
}}
{{eqn | r = 5^{\frac 1 4} \phi^{\frac 3 2}
... | Tangent of 72 Degrees | https://proofwiki.org/wiki/Tangent_of_72_Degrees | https://proofwiki.org/wiki/Tangent_of_72_Degrees | [
"Golden Mean",
"Tangent Function"
] | [
"Definition:Tangent Function"
] | [
"Tangent is Sine divided by Cosine",
"Category:Golden Mean",
"Category:Tangent Function"
] |
proofwiki-23235 | Cotangent of 36 Degrees | :$\cot 36 \degrees = \cot \dfrac \pi 5 = 5^{-\frac 1 4} \phi^{\frac 3 2}$
where $\cot$ denotes the cotangent function. | {{begin-eqn}}
{{eqn | l = \cot 36 \degrees
| r = \frac {\cos 36 \degrees} {\sin 36 \degrees}
| c = Cotangent is Cosine divided by Sine
}}
{{eqn | r = \frac {2^{-1} \phi} {2^{-1} 5^{\frac 1 4} \phi^{-\frac 1 2} }
| c = {{sin|36}} and {{cos|36}}
}}
{{eqn | r = 5^{-\frac 1 4} \phi^{\frac 3 2}
| c =... | :$\cot 36 \degrees = \cot \dfrac \pi 5 = 5^{-\frac 1 4} \phi^{\frac 3 2}$
where $\cot$ denotes the [[Definition:Cotangent|cotangent function]]. | {{begin-eqn}}
{{eqn | l = \cot 36 \degrees
| r = \frac {\cos 36 \degrees} {\sin 36 \degrees}
| c = [[Cotangent is Cosine divided by Sine]]
}}
{{eqn | r = \frac {2^{-1} \phi} {2^{-1} 5^{\frac 1 4} \phi^{-\frac 1 2} }
| c = {{sin|36}} and {{cos|36}}
}}
{{eqn | r = 5^{-\frac 1 4} \phi^{\frac 3 2}
|... | Cotangent of 36 Degrees | https://proofwiki.org/wiki/Cotangent_of_36_Degrees | https://proofwiki.org/wiki/Cotangent_of_36_Degrees | [
"Cotangent Function",
"Golden Mean"
] | [
"Definition:Cotangent"
] | [
"Cotangent is Cosine divided by Sine",
"Category:Cotangent Function",
"Category:Golden Mean"
] |
proofwiki-23236 | Cotangent of 72 Degrees | :$\cot 72 \degrees = \cot \dfrac {2 \pi} 5 = 5^{-\frac 1 4} \phi^{-\frac 3 2}$
where $\cot$ denotes the cotangent function. | {{begin-eqn}}
{{eqn | l = \cot 72 \degrees
| r = \frac {\cos 72 \degrees} {\sin 72 \degrees}
| c = Tangent is Sine divided by Cosine
}}
{{eqn | r = \frac {2^{-1} \phi^{-1} } {2^{-1} 5^{\frac 1 4} \phi^{\frac 1 2} }
| c = {{sin|72}} and {{cos|72}}
}}
{{eqn | r = 5^{-\frac 1 4} \phi^{-\frac 3 2}
|... | :$\cot 72 \degrees = \cot \dfrac {2 \pi} 5 = 5^{-\frac 1 4} \phi^{-\frac 3 2}$
where $\cot$ denotes the [[Definition:Cotangent|cotangent function]]. | {{begin-eqn}}
{{eqn | l = \cot 72 \degrees
| r = \frac {\cos 72 \degrees} {\sin 72 \degrees}
| c = [[Tangent is Sine divided by Cosine]]
}}
{{eqn | r = \frac {2^{-1} \phi^{-1} } {2^{-1} 5^{\frac 1 4} \phi^{\frac 1 2} }
| c = {{sin|72}} and {{cos|72}}
}}
{{eqn | r = 5^{-\frac 1 4} \phi^{-\frac 3 2}
... | Cotangent of 72 Degrees | https://proofwiki.org/wiki/Cotangent_of_72_Degrees | https://proofwiki.org/wiki/Cotangent_of_72_Degrees | [
"Cotangent Function",
"Golden Mean"
] | [
"Definition:Cotangent"
] | [
"Tangent is Sine divided by Cosine",
"Category:Cotangent Function",
"Category:Golden Mean"
] |
proofwiki-23237 | Digamma Function of One Fifth | :$\map \psi {\dfrac 1 5} = -\gamma - \dfrac 5 4 \ln 5 - \dfrac \pi 2 5^{-\frac 1 4} \phi^{\frac 3 2} + \paren {\dfrac 1 2 - \phi} \ln \phi$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 5}
| r = -\gamma - \ln 10 - \frac \pi 2 \map \cot {\frac 1 5 \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {5 / 2} - 1} \map \cos {\frac {2 \pi n} 5} \map \ln {\map \sin {\frac {\pi n} 5} }
| c = Gauss's Digamma Theorem
}}
{{eqn | r = -\gamma - \ln 2 - \ln 5 - \frac \p... | :$\map \psi {\dfrac 1 5} = -\gamma - \dfrac 5 4 \ln 5 - \dfrac \pi 2 5^{-\frac 1 4} \phi^{\frac 3 2} + \paren {\dfrac 1 2 - \phi} \ln \phi$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 5}
| r = -\gamma - \ln 10 - \frac \pi 2 \map \cot {\frac 1 5 \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {5 / 2} - 1} \map \cos {\frac {2 \pi n} 5} \map \ln {\map \sin {\frac {\pi n} 5} }
| c = [[Gauss's Digamma Theorem]]
}}
{{eqn | r = -\gamma - \ln 2 - \ln 5 - \fra... | Digamma Function of One Fifth | https://proofwiki.org/wiki/Digamma_Function_of_One_Fifth | https://proofwiki.org/wiki/Digamma_Function_of_One_Fifth | [
"Digamma Function of One Fifth",
"Examples of Digamma Function",
"Euler-Mascheroni Constant"
] | [] | [
"Gauss's Digamma Theorem",
"Sum of Logarithms",
"Sum of Logarithms",
"Category:Digamma Function of One Fifth",
"Category:Examples of Digamma Function",
"Category:Euler-Mascheroni Constant"
] |
proofwiki-23238 | Digamma Function of Two Fifths | :$\map \psi {\dfrac 2 5} = -\gamma - \dfrac 5 4 \ln 5 - \dfrac \pi 2 5^{-\frac 1 4} \phi^{-\frac 3 2} + \paren {\phi - \dfrac 1 2} \ln \phi$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 2 5}
| r = -\gamma - \ln 10 - \frac \pi 2 \map \cot {\frac 2 5 \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {5 / 2} - 1} \map \cos {\frac {4 \pi n} 5} \map \ln {\map \sin {\frac {\pi n} 5} }
| c = Gauss's Digamma Theorem
}}
{{eqn | r = -\gamma - \ln 2 - \ln 5 - \frac \p... | :$\map \psi {\dfrac 2 5} = -\gamma - \dfrac 5 4 \ln 5 - \dfrac \pi 2 5^{-\frac 1 4} \phi^{-\frac 3 2} + \paren {\phi - \dfrac 1 2} \ln \phi$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 2 5}
| r = -\gamma - \ln 10 - \frac \pi 2 \map \cot {\frac 2 5 \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {5 / 2} - 1} \map \cos {\frac {4 \pi n} 5} \map \ln {\map \sin {\frac {\pi n} 5} }
| c = [[Gauss's Digamma Theorem]]
}}
{{eqn | r = -\gamma - \ln 2 - \ln 5 - \fra... | Digamma Function of Two Fifths | https://proofwiki.org/wiki/Digamma_Function_of_Two_Fifths | https://proofwiki.org/wiki/Digamma_Function_of_Two_Fifths | [
"Examples of Digamma Function",
"Euler-Mascheroni Constant"
] | [] | [
"Gauss's Digamma Theorem",
"Cosine Function is Even",
"Sum of Logarithms",
"Sum of Logarithms",
"Category:Examples of Digamma Function",
"Category:Euler-Mascheroni Constant"
] |
proofwiki-23239 | Digamma Function of Three Fifths | :$\map \psi {\dfrac 3 5} = -\gamma - \dfrac 5 4 \ln 5 + \dfrac \pi 2 5^{-\frac 1 4} \phi^{-\frac 3 2} + \paren {\phi - \dfrac 1 2} \ln \phi$ | From Digamma Reflection Formula. we have:
:$\map \psi z - \map \psi {1 - z} = -\pi \cot \pi z$
Therefore:
{{begin-eqn}}
{{eqn | l = \map \psi {\frac 3 5} - \map \psi {\frac 2 5}
| r = -\pi \map \cot {\frac {3 \pi} 5}
| c =
}}
{{eqn | ll = \leadsto
| l = \map \psi {\frac 3 5}
| r = \map \psi {\f... | :$\map \psi {\dfrac 3 5} = -\gamma - \dfrac 5 4 \ln 5 + \dfrac \pi 2 5^{-\frac 1 4} \phi^{-\frac 3 2} + \paren {\phi - \dfrac 1 2} \ln \phi$ | From [[Digamma Reflection Formula]]. we have:
:$\map \psi z - \map \psi {1 - z} = -\pi \cot \pi z$
Therefore:
{{begin-eqn}}
{{eqn | l = \map \psi {\frac 3 5} - \map \psi {\frac 2 5}
| r = -\pi \map \cot {\frac {3 \pi} 5}
| c =
}}
{{eqn | ll = \leadsto
| l = \map \psi {\frac 3 5}
| r = \map \p... | Digamma Function of Three Fifths | https://proofwiki.org/wiki/Digamma_Function_of_Three_Fifths | https://proofwiki.org/wiki/Digamma_Function_of_Three_Fifths | [
"Examples of Digamma Function",
"Euler-Mascheroni Constant"
] | [] | [
"Digamma Reflection Formula",
"Digamma Function of Two Fifths",
"Cotangent Function is Odd",
"Cotangent of Angle plus Straight Angle",
"Category:Examples of Digamma Function",
"Category:Euler-Mascheroni Constant"
] |
proofwiki-23240 | Digamma Function of Four Fifths | :$\map \psi {\dfrac 4 5} = -\gamma - \dfrac 5 4 \ln 5 + \dfrac \pi 2 5^{-\frac 1 4} \phi^{\frac 3 2} + \paren {\dfrac 1 2 - \phi} \ln \phi$ | From Digamma Reflection Formula. we have:
:$\map \psi z - \map \psi {1 - z} = -\pi \cot \pi z$
Therefore:
{{begin-eqn}}
{{eqn | l = \map \psi {\frac 4 5} - \map \psi {\frac 1 5}
| r = -\pi \map \cot {\frac {4 \pi} 5}
| c =
}}
{{eqn | ll = \leadsto
| l = \map \psi {\frac 4 5}
| r = \map \psi {\f... | :$\map \psi {\dfrac 4 5} = -\gamma - \dfrac 5 4 \ln 5 + \dfrac \pi 2 5^{-\frac 1 4} \phi^{\frac 3 2} + \paren {\dfrac 1 2 - \phi} \ln \phi$ | From [[Digamma Reflection Formula]]. we have:
:$\map \psi z - \map \psi {1 - z} = -\pi \cot \pi z$
Therefore:
{{begin-eqn}}
{{eqn | l = \map \psi {\frac 4 5} - \map \psi {\frac 1 5}
| r = -\pi \map \cot {\frac {4 \pi} 5}
| c =
}}
{{eqn | ll = \leadsto
| l = \map \psi {\frac 4 5}
| r = \map \p... | Digamma Function of Four Fifths | https://proofwiki.org/wiki/Digamma_Function_of_Four_Fifths | https://proofwiki.org/wiki/Digamma_Function_of_Four_Fifths | [
"Examples of Digamma Function",
"Euler-Mascheroni Constant"
] | [] | [
"Digamma Reflection Formula",
"Digamma Function of One Fifth",
"Cotangent Function is Odd",
"Cotangent of Angle plus Straight Angle",
"Category:Examples of Digamma Function",
"Category:Euler-Mascheroni Constant"
] |
proofwiki-23241 | Product Space is T0 iff Factor Spaces are T0/Necessary Condition | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces where $I$ is an arbitrary index set.
Let $\ds T = \struct {S, \tau} = \prod \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$.
Let each of $\struct {S_\alpha, \tau_\alpha}$ be ... | Let each of $\struct {S_\alpha, \tau_\alpha}$ be a $T_0$ space.
{{Recall|T0 Space|$T_0$ space|1}}
{{:Definition:T0 Space/Definition 1}}
{{AimForCont}} $T$ is not a $T_0$ space.
Then $\exists a, b \in S, a \ne b$ such that for all $U \in \tau$, either $a, b \in U$ or $a, b \notin U$.
Then $a$ and $b$ are different in at... | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological spaces]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $\ds T = \struct {S, \tau} ... | Let each of $\struct {S_\alpha, \tau_\alpha}$ be a [[Definition:T0 Space|$T_0$ space]].
{{Recall|T0 Space|$T_0$ space|1}}
{{:Definition:T0 Space/Definition 1}}
{{AimForCont}} $T$ is not a [[Definition:T0 Space|$T_0$ space]].
Then $\exists a, b \in S, a \ne b$ such that for all $U \in \tau$, either $a, b \in U$ or $a... | Product Space is T0 iff Factor Spaces are T0/Necessary Condition | https://proofwiki.org/wiki/Product_Space_is_T0_iff_Factor_Spaces_are_T0/Necessary_Condition | https://proofwiki.org/wiki/Product_Space_is_T0_iff_Factor_Spaces_are_T0/Necessary_Condition | [
"Product Space is T0 iff Factor Spaces are T0"
] | [
"Definition:Indexing Set/Family",
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:T0 Space",
"Definition:T0 Space"
] | [
"Definition:T0 Space",
"Definition:T0 Space",
"Definition:Cartesian Product/Coordinate",
"Definition:Cartesian Product/Coordinate",
"Definition:T0 Space",
"Proof by Contradiction",
"Definition:T0 Space"
] |
proofwiki-23242 | Product Space is T0 iff Factor Spaces are T0/Sufficient Condition | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces where $I$ is an arbitrary index set.
Let $\ds T = \struct {S, \tau} = \prod \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$.
Let $T$ be a $T_0$ space.
Then each of $\struct {... | Let $T$ be a $T_0$ space.
As $S_\alpha \ne \O$ we also have $S \ne \O$ by the axiom of choice.
Let $\alpha \in I$.
From Subspace of Product Space is Homeomorphic to Factor Space, $\struct {S_\alpha, \tau_\alpha}$ is homeomorphic to a subspace $T_\alpha$ of $T$.
From $T_0$ Property is Hereditary, $T_\alpha$ is a $T_0$ s... | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological spaces]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $\ds T = \struct {S, \tau} ... | Let $T$ be a [[Definition:T0 Space|$T_0$ space]].
As $S_\alpha \ne \O$ we also have $S \ne \O$ by the [[Axiom:Axiom of Choice|axiom of choice]].
Let $\alpha \in I$.
From [[Subspace of Product Space is Homeomorphic to Factor Space]], $\struct {S_\alpha, \tau_\alpha}$ is [[Definition:Homeomorphism (Topological Spaces)... | Product Space is T0 iff Factor Spaces are T0/Sufficient Condition/Proof 1 | https://proofwiki.org/wiki/Product_Space_is_T0_iff_Factor_Spaces_are_T0/Sufficient_Condition | https://proofwiki.org/wiki/Product_Space_is_T0_iff_Factor_Spaces_are_T0/Sufficient_Condition/Proof_1 | [
"Product Space is T0 iff Factor Spaces are T0"
] | [
"Definition:Indexing Set/Family",
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:T0 Space",
"Definition:T0 Space"
] | [
"Definition:T0 Space",
"Axiom:Axiom of Choice",
"Subspace of Product Space is Homeomorphic to Factor Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Subspace",
"T0 Property is Hereditary",
"Definition:T0 Space",
"T0 Property is Preserved under Homeomorphism",
"Definition:T0 Space"... |
proofwiki-23243 | Product Space is T0 iff Factor Spaces are T0/Sufficient Condition | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces where $I$ is an arbitrary index set.
Let $\ds T = \struct {S, \tau} = \prod \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$.
Let $T$ be a $T_0$ space.
Then each of $\struct {... | Let $T$ be a $T_0$ space.
{{AimForCont}} $\exists \beta: \struct {S_\beta, \tau_\beta}$ is not a $T_0$ space.
Then $\exists a, b \in S_\beta$ such that $\forall U_\beta \in \tau_\beta$, either $a, b \in U_\beta$ or $a, b \notin U_\beta$.
Consider the elements $y, z \in S$ defined as:
:<nowiki>$y = \family {x_\alpha} :... | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological spaces]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $\ds T = \struct {S, \tau} ... | Let $T$ be a [[Definition:T0 Space|$T_0$ space]].
{{AimForCont}} $\exists \beta: \struct {S_\beta, \tau_\beta}$ is not a [[Definition:T0 Space|$T_0$ space]].
Then $\exists a, b \in S_\beta$ such that $\forall U_\beta \in \tau_\beta$, either $a, b \in U_\beta$ or $a, b \notin U_\beta$.
Consider the [[Definition:Eleme... | Product Space is T0 iff Factor Spaces are T0/Sufficient Condition/Proof 2 | https://proofwiki.org/wiki/Product_Space_is_T0_iff_Factor_Spaces_are_T0/Sufficient_Condition | https://proofwiki.org/wiki/Product_Space_is_T0_iff_Factor_Spaces_are_T0/Sufficient_Condition/Proof_2 | [
"Product Space is T0 iff Factor Spaces are T0"
] | [
"Definition:Indexing Set/Family",
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:T0 Space",
"Definition:T0 Space"
] | [
"Definition:T0 Space",
"Definition:T0 Space",
"Definition:Element",
"Definition:Cartesian Product/Coordinate",
"Definition:T0 Space",
"Proof by Contradiction",
"Definition:T0 Space"
] |
proofwiki-23244 | Product Space is T1 iff Factor Spaces are T1/Sufficient Condition | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces where $I$ is an arbitrary index set.
Let $\ds T = \struct {S, \tau} = \prod \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$.
Let $T$ be a $T_1$ space.
Then each of $\struct {... | Let $T$ be a $T_1$ space.
As $S_\alpha \ne \O$ we also have $S \ne \O$ by the axiom of choice.
From Subspace of Product Space is Homeomorphic to Factor Space, $\struct {S_\alpha, \tau_\alpha}$ is homeomorphic to a subspace $T_\alpha$ of $T$.
From $T_1$ property is hereditary, $T_\alpha$ is a $T_1$ space.
From $T_1$ Pro... | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological spaces]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $\ds T = \struct {S, \tau} ... | Let $T$ be a [[Definition:T1 Space|$T_1$ space]].
As $S_\alpha \ne \O$ we also have $S \ne \O$ by the [[Axiom:Axiom of Choice|axiom of choice]].
From [[Subspace of Product Space is Homeomorphic to Factor Space]], $\struct {S_\alpha, \tau_\alpha}$ is [[Definition:Homeomorphism (Topological Spaces)|homeomorphic]] to a ... | Product Space is T1 iff Factor Spaces are T1/Sufficient Condition/Proof 1 | https://proofwiki.org/wiki/Product_Space_is_T1_iff_Factor_Spaces_are_T1/Sufficient_Condition | https://proofwiki.org/wiki/Product_Space_is_T1_iff_Factor_Spaces_are_T1/Sufficient_Condition/Proof_1 | [
"Product Space is T1 iff Factor Spaces are T1"
] | [
"Definition:Indexing Set/Family",
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:T1 Space",
"Definition:T1 Space"
] | [
"Definition:T1 Space",
"Axiom:Axiom of Choice",
"Subspace of Product Space is Homeomorphic to Factor Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Subspace",
"T1 Property is Hereditary",
"Definition:T1 Space",
"T1 Property is Preserved under Homeomorphism",
"Definition:T1 Space"... |
proofwiki-23245 | Product Space is T1 iff Factor Spaces are T1/Necessary Condition | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces where $I$ is an arbitrary index set.
Let $\ds T = \struct {S, \tau} = \prod \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$.
Let each of $\struct {S_\alpha, \tau_\alpha}$ be ... | Let each of $\struct {S_\alpha, \tau_\alpha}$ be a $T_1$ space.
{{Recall|T1 Space|$T_1$ space|1}}
{{:Definition:T1 Space/Definition 1}}
{{AimForCont}} $T$ is not a $T_1$ space.
Then $\exists a, b \in S, a \ne b$ such that for all $U \in \tau$, $a \in U \implies b \in U$.
Then $a$ and $b$ are different in at least one c... | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological spaces]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $\ds T = \struct {S, \tau} ... | Let each of $\struct {S_\alpha, \tau_\alpha}$ be a [[Definition:T1 Space|$T_1$ space]].
{{Recall|T1 Space|$T_1$ space|1}}
{{:Definition:T1 Space/Definition 1}}
{{AimForCont}} $T$ is not a [[Definition:T1 Space|$T_1$ space]].
Then $\exists a, b \in S, a \ne b$ such that for all $U \in \tau$, $a \in U \implies b \in U... | Product Space is T1 iff Factor Spaces are T1/Necessary Condition | https://proofwiki.org/wiki/Product_Space_is_T1_iff_Factor_Spaces_are_T1/Necessary_Condition | https://proofwiki.org/wiki/Product_Space_is_T1_iff_Factor_Spaces_are_T1/Necessary_Condition | [
"Product Space is T1 iff Factor Spaces are T1"
] | [
"Definition:Indexing Set/Family",
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:T1 Space",
"Definition:T0 Space"
] | [
"Definition:T1 Space",
"Definition:T1 Space",
"Definition:T1 Space",
"Proof by Contradiction",
"Definition:T0 Space"
] |
proofwiki-23246 | Product Space is T2 iff Factor Spaces are T2/Sufficient Condition | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces where $I$ is an arbitrary index set.
Let $\ds T = \struct {S, \tau} = \prod \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$.
Let $T$ be a $T_2$ space.
Then each of $\struct {... | Let $T$ be a $T_2$ space.
As $S_\alpha \ne \O$ we also have $S \ne \O$ by the axiom of choice.
From Subspace of Product Space is Homeomorphic to Factor Space, $\struct {S_\alpha, \tau_\alpha}$ is homeomorphic to a subspace $T_\alpha$ of $T$.
From $T_2$ property is hereditary, $T_\alpha$ is a $T_2$ space.
From $T_2$ Pro... | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological spaces]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $\ds T = \struct {S, \tau} ... | Let $T$ be a [[Definition:T2 Space|$T_2$ space]].
As $S_\alpha \ne \O$ we also have $S \ne \O$ by the [[Axiom:Axiom of Choice|axiom of choice]].
From [[Subspace of Product Space is Homeomorphic to Factor Space]], $\struct {S_\alpha, \tau_\alpha}$ is [[Definition:Homeomorphism (Topological Spaces)|homeomorphic]] to a ... | Product Space is T2 iff Factor Spaces are T2/Sufficient Condition/Proof 1 | https://proofwiki.org/wiki/Product_Space_is_T2_iff_Factor_Spaces_are_T2/Sufficient_Condition | https://proofwiki.org/wiki/Product_Space_is_T2_iff_Factor_Spaces_are_T2/Sufficient_Condition/Proof_1 | [
"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",
"Axiom:Axiom of Choice",
"Subspace of Product Space is Homeomorphic to Factor Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Subspace",
"T2 Property is Hereditary",
"Definition:T2 Space",
"T2 Property is Preserved under Homeomorphism",
"Definition:T2 Space"... |
proofwiki-23247 | Product Space is T2 iff Factor Spaces are T2/Sufficient Condition | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces where $I$ is an arbitrary index set.
Let $\ds T = \struct {S, \tau} = \prod \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$.
Let $T$ be a $T_2$ space.
Then each of $\struct {... | Let $\alpha \in I$.
Let $x, y \in S_\alpha$.
By the Axiom of Choice, there exists $z \in S$.
Define $x', y' \in S$ by:
:$x'_\beta = \begin{cases} z_\beta & \beta \ne \alpha \\ x & \beta = \alpha \end{cases}$
and
:$y'_\beta = \begin{cases} z_\beta & \beta \ne \alpha \\ y & \beta = \alpha \end{cases}$
By definition of a ... | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological spaces]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $\ds T = \struct {S, \tau} ... | Let $\alpha \in I$.
Let $x, y \in S_\alpha$.
By the [[Axiom:Axiom of Choice|Axiom of Choice]], there exists $z \in S$.
Define $x', y' \in S$ by:
:$x'_\beta = \begin{cases} z_\beta & \beta \ne \alpha \\ x & \beta = \alpha \end{cases}$
and
:$y'_\beta = \begin{cases} z_\beta & \beta \ne \alpha \\ y & \beta = \alpha \en... | Product Space is T2 iff Factor Spaces are T2/Sufficient Condition/Proof 2 | https://proofwiki.org/wiki/Product_Space_is_T2_iff_Factor_Spaces_are_T2/Sufficient_Condition | https://proofwiki.org/wiki/Product_Space_is_T2_iff_Factor_Spaces_are_T2/Sufficient_Condition/Proof_2 | [
"Product Space is T2 iff Factor Spaces are T2"
] | [
"Definition:Indexing Set/Family",
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:T2 Space",
"Definition:T2 Space"
] | [
"Axiom:Axiom of Choice",
"Definition:T2 Space",
"Definition:Topology",
"Definition:Product Space (Topology)",
"Definition:Product Topology",
"Definition:Product Topology",
"Definition:Product Topology/Natural Basis",
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Product Topology",
"De... |
proofwiki-23248 | Product Space is Regular iff Factor Spaces are Regular | :$T$ is a regular space {{iff}} each of $\struct {S_\alpha, \tau_\alpha}$ is a regular space. | {{Recall|Regular Space|index = 1}}
{{:Definition:Regular Space/Definition 1}}
The result follows from:
:Product Space is $T_0$ iff Factor Spaces are $T_0$
:Product Space is $T_3$ iff Factor Spaces are $T_3$
{{qed}} | :$T$ is a [[Definition:Regular Space|regular space]] {{iff}} each of $\struct {S_\alpha, \tau_\alpha}$ is a [[Definition:Regular Space|regular space]]. | {{Recall|Regular Space|index = 1}}
{{:Definition:Regular Space/Definition 1}}
The result follows from:
:[[Product Space is T0 iff Factor Spaces are T0|Product Space is $T_0$ iff Factor Spaces are $T_0$]]
:[[Product Space is T3 iff Factor Spaces are T3|Product Space is $T_3$ iff Factor Spaces are $T_3$]]
{{qed}} | Product Space is Regular iff Factor Spaces are Regular | https://proofwiki.org/wiki/Product_Space_is_Regular_iff_Factor_Spaces_are_Regular | https://proofwiki.org/wiki/Product_Space_is_Regular_iff_Factor_Spaces_are_Regular | [
"Regular Spaces",
"Separation Properties Preserved under Topological Product",
"Product Spaces"
] | [
"Definition:Regular Space",
"Definition:Regular Space"
] | [
"Product Space is T0 iff Factor Spaces are T0",
"Product Space is T3 iff Factor Spaces are T3"
] |
proofwiki-23249 | Product Space is Completely Regular iff Factor Spaces are Completely Regular | :$T$ is a completely regular space {{iff}} each of $\struct {S_\alpha, \tau_\alpha}$ is a completely regular space. | {{Recall|Completely Regular Space|completely regular space|index = 1}}
{{:Definition:Completely Regular Space/Definition 1}}
The result follows from:
:Product Space is $T_0$ iff Factor Spaces are $T_0$
:Product Space is $T_{3 \frac 1 2}$ iff Factor Spaces are $T_{3 \frac 1 2}$
{{qed}} | :$T$ is a [[Definition:Completely Regular Space|completely regular space]] {{iff}} each of $\struct {S_\alpha, \tau_\alpha}$ is a [[Definition:Completely Regular Space|completely regular space]]. | {{Recall|Completely Regular Space|completely regular space|index = 1}}
{{:Definition:Completely Regular Space/Definition 1}}
The result follows from:
:[[Product Space is T0 iff Factor Spaces are T0|Product Space is $T_0$ iff Factor Spaces are $T_0$]]
:[[Product Space is T3.5 iff Factor Spaces are T3.5|Product Space is... | Product Space is Completely Regular iff Factor Spaces are Completely Regular | https://proofwiki.org/wiki/Product_Space_is_Completely_Regular_iff_Factor_Spaces_are_Completely_Regular | https://proofwiki.org/wiki/Product_Space_is_Completely_Regular_iff_Factor_Spaces_are_Completely_Regular | [
"Completely Regular Spaces",
"Separation Properties Preserved under Topological Product",
"Product Spaces"
] | [
"Definition:Completely Regular Space",
"Definition:Completely Regular Space"
] | [
"Product Space is T0 iff Factor Spaces are T0",
"Product Space is T3.5 iff Factor Spaces are T3.5"
] |
proofwiki-23250 | Factor Spaces are Normal if Product Space is Normal | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces where $I$ is an arbitrary index set.
Let $\ds T = \struct {S, \tau} = \prod \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$.
Let $T$ be a normal space.
Then each of $\struct ... | {{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
The result follows from:
:Product Space is $T_1$ iff Factor Spaces are $T_1$
:Factor Spaces are $T_4$ if Product Space is $T_4$
{{qed}} | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological spaces]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $\ds T = \struct {S, \tau} ... | {{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
The result follows from:
:[[Product Space is T1 iff Factor Spaces are T1|Product Space is $T_1$ iff Factor Spaces are $T_1$]]
:[[Factor Spaces are T4 if Product Space is T4|Factor Spaces are $T_4$ if Product Space is $T_4$]]
{{qed... | Factor Spaces are Normal if Product Space is Normal | https://proofwiki.org/wiki/Factor_Spaces_are_Normal_if_Product_Space_is_Normal | https://proofwiki.org/wiki/Factor_Spaces_are_Normal_if_Product_Space_is_Normal | [
"Normal Spaces",
"Separation Properties Preserved under Topological Product",
"Product Spaces"
] | [
"Definition:Indexing Set/Family",
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:Normal Space",
"Definition:Normal Space"
] | [
"Product Space is T1 iff Factor Spaces are T1",
"Factor Spaces are T4 if Product Space is T4"
] |
proofwiki-23251 | Factor Spaces are Completely Normal if Product Space is Completely Normal | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces where $I$ is an arbitrary index set.
Let $\ds T = \struct {S, \tau} = \prod \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$.
Let $T$ be a completely normal space.
Then each o... | {{Recall|Completely Normal Space|completely normal space|index = 1}}
{{:Definition:Completely Normal Space/Definition 1}}
The result follows from:
{{begin-itemize}}
{{item||Product Space is $T_1$ iff Factor Spaces are $T_1$}}
{{item||Factor Spaces are $T_5$ if Product Space is $T_5$.}}
{{end-itemize}}
{{qed}} | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological spaces]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $\ds T = \struct {S, \tau} ... | {{Recall|Completely Normal Space|completely normal space|index = 1}}
{{:Definition:Completely Normal Space/Definition 1}}
The result follows from:
{{begin-itemize}}
{{item||[[Product Space is T1 iff Factor Spaces are T1|Product Space is $T_1$ iff Factor Spaces are $T_1$]]}}
{{item||[[Factor Spaces are T5 if Product Sp... | Factor Spaces are Completely Normal if Product Space is Completely Normal | https://proofwiki.org/wiki/Factor_Spaces_are_Completely_Normal_if_Product_Space_is_Completely_Normal | https://proofwiki.org/wiki/Factor_Spaces_are_Completely_Normal_if_Product_Space_is_Completely_Normal | [
"Completely Normal Spaces",
"Separation Properties Preserved under Topological Product",
"Product Spaces"
] | [
"Definition:Indexing Set/Family",
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:Completely Normal Space",
"Definition:Completely Normal Space"
] | [
"Product Space is T1 iff Factor Spaces are T1",
"Factor Spaces are T5 if Product Space is T5"
] |
proofwiki-23252 | Product of Normal Factor Spaces is not necessarily Normal | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces where $I$ is an arbitrary index set.
Let $\ds T = \struct {S, \tau} = \prod \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$.
Let each of $\struct {S_\alpha, \tau_\alpha}$ be ... | From Existence of Completely Normal Space whose Product Space is Not Normal, there exist completely normal topological spaces $T_A$ and $T_B$ such that their product space $T_A \times T_B$ is not a normal space.
From Completely Normal Space is Normal, $T_A$ and $T_B$ are both normal spaces.
Hence there exist normal spa... | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological spaces]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $\ds T = \struct {S, \tau} ... | From [[Existence of Completely Normal Space whose Product Space is Not Normal]], there exist [[Definition:Completely Normal Space|completely normal]] [[Definition:Topological Space|topological spaces]] $T_A$ and $T_B$ such that their [[Definition:Product Space of Topological Spaces|product space]] $T_A \times T_B$ is n... | Product of Normal Factor Spaces is not necessarily Normal | https://proofwiki.org/wiki/Product_of_Normal_Factor_Spaces_is_not_necessarily_Normal | https://proofwiki.org/wiki/Product_of_Normal_Factor_Spaces_is_not_necessarily_Normal | [
"Normal Spaces",
"Separation Properties Preserved under Topological Product",
"Product Spaces"
] | [
"Definition:Indexing Set/Family",
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:Normal Space",
"Definition:Normal Space"
] | [
"Existence of Completely Normal Space whose Product Space is Not Normal",
"Definition:Completely Normal Space",
"Definition:Topological Space",
"Definition:Product Space (Topology)",
"Definition:Normal Space",
"Completely Normal Space is Normal",
"Definition:Normal Space",
"Definition:Normal Space",
... |
proofwiki-23253 | Product of Completely Normal Factor Spaces is not necessarily Completely Normal | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces where $I$ is an arbitrary index set.
Let $\ds T = \struct {S, \tau} = \prod \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$.
Let each of $\struct {S_\alpha, \tau_\alpha}$ be ... | From Existence of Completely Normal Space whose Product Space is Not Normal, there exist completely normal topological spaces $T_A$ and $T_B$ such that their product space $T = T_A \times T_B$ is not a normal space.
As $T$ is not a normal space, it follows from Completely Normal Space is Normal that $T$ is not a comple... | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological spaces]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $\ds T = \struct {S, \tau} ... | From [[Existence of Completely Normal Space whose Product Space is Not Normal]], there exist [[Definition:Completely Normal Space|completely normal]] [[Definition:Topological Space|topological spaces]] $T_A$ and $T_B$ such that their [[Definition:Product Space of Topological Spaces|product space]] $T = T_A \times T_B$ ... | Product of Completely Normal Factor Spaces is not necessarily Completely Normal | https://proofwiki.org/wiki/Product_of_Completely_Normal_Factor_Spaces_is_not_necessarily_Completely_Normal | https://proofwiki.org/wiki/Product_of_Completely_Normal_Factor_Spaces_is_not_necessarily_Completely_Normal | [
"Completely Normal Spaces",
"Separation Properties Preserved under Topological Product",
"Product Spaces"
] | [
"Definition:Indexing Set/Family",
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:Completely Normal Space",
"Definition:Completely Normal Space"
] | [
"Existence of Completely Normal Space whose Product Space is Not Normal",
"Definition:Completely Normal Space",
"Definition:Topological Space",
"Definition:Product Space (Topology)",
"Definition:Normal Space",
"Definition:Normal Space",
"Completely Normal Space is Normal",
"Definition:Completely Norma... |
proofwiki-23254 | Product of T4 Factor Spaces is not necessarily T4 | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces where $I$ is an arbitrary index set.
Let $\ds T = \struct {S, \tau} = \prod \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$.
Let each of $\struct {S_\alpha, \tau_\alpha}$ be ... | From Product of Normal Factor Spaces is not necessarily Normal, there exist normal topological spaces $T_A$ and $T_B$ such that their product space $T = T_A \times T_B$ is not a normal space.
{{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
As $T_A$ and $T_B$ are both normal spac... | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological spaces]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $\ds T = \struct {S, \tau} ... | From [[Product of Normal Factor Spaces is not necessarily Normal]], there exist [[Definition:Normal Space|normal]] [[Definition:Topological Space|topological spaces]] $T_A$ and $T_B$ such that their [[Definition:Product Space of Topological Spaces|product space]] $T = T_A \times T_B$ is not a [[Definition:Normal Space|... | Product of T4 Factor Spaces is not necessarily T4 | https://proofwiki.org/wiki/Product_of_T4_Factor_Spaces_is_not_necessarily_T4 | https://proofwiki.org/wiki/Product_of_T4_Factor_Spaces_is_not_necessarily_T4 | [
"T4 Spaces",
"Separation Properties Preserved under Topological Product",
"Product Spaces"
] | [
"Definition:Indexing Set/Family",
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:T4 Space",
"Definition:T4 Space"
] | [
"Product of Normal Factor Spaces is not necessarily Normal",
"Definition:Normal Space",
"Definition:Topological Space",
"Definition:Product Space (Topology)",
"Definition:Normal Space",
"Definition:Normal Space",
"Definition:T1 Space",
"Product Space is T1 iff Factor Spaces are T1",
"Definition:Prod... |
proofwiki-23255 | Product of T5 Factor Spaces is not necessarily T5 | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces where $I$ is an arbitrary index set.
Let $\ds T = \struct {S, \tau} = \prod \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$.
Let each of $\struct {S_\alpha, \tau_\alpha}$ be ... | From Product of Completely Normal Factor Spaces is not necessarily Completely Normal, there exist normal topological spaces $T_A$ and $T_B$ such that their product space $T = T_A \times T_B$ is not a completely normal space.
Let $T = T_A \times T_B$ be a product space such that:
{{begin-itemize}}
{{item||$T_A$ and $T_B... | Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Topological Space|topological spaces]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $\ds T = \struct {S, \tau} ... | From [[Product of Completely Normal Factor Spaces is not necessarily Completely Normal]], there exist [[Definition:Normal Space|normal]] [[Definition:Topological Space|topological spaces]] $T_A$ and $T_B$ such that their [[Definition:Product Space of Topological Spaces|product space]] $T = T_A \times T_B$ is not a [[De... | Product of T5 Factor Spaces is not necessarily T5 | https://proofwiki.org/wiki/Product_of_T5_Factor_Spaces_is_not_necessarily_T5 | https://proofwiki.org/wiki/Product_of_T5_Factor_Spaces_is_not_necessarily_T5 | [
"T5 Spaces",
"Separation Properties Preserved under Topological Product",
"Product Spaces"
] | [
"Definition:Indexing Set/Family",
"Definition:Non-Empty Set",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Product Space (Topology)",
"Definition:T5 Space",
"Definition:T5 Space"
] | [
"Product of Completely Normal Factor Spaces is not necessarily Completely Normal",
"Definition:Normal Space",
"Definition:Topological Space",
"Definition:Product Space (Topology)",
"Definition:Completely Normal Space",
"Definition:Product Space (Topology)",
"Definition:Normal Space",
"Definition:Compl... |
proofwiki-23256 | T0 Property is Preserved under Expansion | If $T_A$ is a $T_0$ space, then so is $T_B$. | Let $T_A = \struct {S, \tau_1}$ be a $T_0$ space.
Let $T_B = \struct {S, \tau_2}$ be such that $\tau_2$ is an expansion of $\tau_1$.
From Identity Mapping to Expansion is Closed, we have that $I_S$ is closed.
We also have Identity Mapping is Bijection.
Hence we can directly apply:
:$T_0$ Property is Preserved under Clo... | If $T_A$ is a [[Definition:T0 Space|$T_0$ space]], then so is $T_B$. | Let $T_A = \struct {S, \tau_1}$ be a [[Definition:T0 Space|$T_0$ space]].
Let $T_B = \struct {S, \tau_2}$ be such that $\tau_2$ is an [[Definition:Expansion of Topology|expansion]] of $\tau_1$.
From [[Identity Mapping to Expansion is Closed]], we have that $I_S$ is [[Definition:Closed Mapping|closed]].
We also have... | T0 Property is Preserved under Expansion | https://proofwiki.org/wiki/T0_Property_is_Preserved_under_Expansion | https://proofwiki.org/wiki/T0_Property_is_Preserved_under_Expansion | [
"T0 Spaces",
"Separation Properties Preserved under Expansion"
] | [
"Definition:T0 Space"
] | [
"Definition:T0 Space",
"Definition:Expansion of Topology",
"Identity Mapping to Expansion is Closed",
"Definition:Closed Mapping",
"Identity Mapping is Bijection",
"T0 Property is Preserved under Closed Bijection",
"Definition:T0 Space",
"Definition:T0 Space"
] |
proofwiki-23257 | T1 Property is Preserved under Expansion | If $T_A$ is a $T_1$ space, then so is $T_B$. | Let $T_A = \struct {S, \tau_1}$ be a $T_1$ space.
Let $T_B = \struct {S, \tau_2}$ be such that $\tau_2$ is an expansion of $\tau_1$.
From Identity Mapping to Expansion is Closed, we have that $I_S$ is closed.
We also have Identity Mapping is Bijection.
Hence we can directly apply:
:$T_1$ Property is Preserved under Clo... | If $T_A$ is a [[Definition:T1 Space|$T_1$ space]], then so is $T_B$. | Let $T_A = \struct {S, \tau_1}$ be a [[Definition:T1 Space|$T_1$ space]].
Let $T_B = \struct {S, \tau_2}$ be such that $\tau_2$ is an [[Definition:Expansion of Topology|expansion]] of $\tau_1$.
From [[Identity Mapping to Expansion is Closed]], we have that $I_S$ is [[Definition:Closed Mapping|closed]].
We also have... | T1 Property is Preserved under Expansion | https://proofwiki.org/wiki/T1_Property_is_Preserved_under_Expansion | https://proofwiki.org/wiki/T1_Property_is_Preserved_under_Expansion | [
"T1 Spaces",
"Separation Properties Preserved under Expansion"
] | [
"Definition:T1 Space"
] | [
"Definition:T1 Space",
"Definition:Expansion of Topology",
"Identity Mapping to Expansion is Closed",
"Definition:Closed Mapping",
"Identity Mapping is Bijection",
"T1 Property is Preserved under Closed Bijection",
"Definition:T1 Space",
"Definition:T1 Space"
] |
proofwiki-23258 | T2 Property is Preserved under Expansion | If $T_A$ is a $T_2$ (Hausdorff) space, then so is $T_B$. | Let $T_A = \struct {S, \tau_1}$ be a $T_2$ space.
Let $T_B = \struct {S, \tau_2}$ be such that $\tau_2$ is an expansion of $\tau_1$.
From Identity Mapping to Expansion is Closed, we have that $I_S$ is closed.
We also have Identity Mapping is Bijection.
Hence we can directly apply:
:$T_2$ Property is Preserved under Clo... | If $T_A$ is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]], then so is $T_B$. | Let $T_A = \struct {S, \tau_1}$ be a [[Definition:T2 Space|$T_2$ space]].
Let $T_B = \struct {S, \tau_2}$ be such that $\tau_2$ is an [[Definition:Expansion of Topology|expansion]] of $\tau_1$.
From [[Identity Mapping to Expansion is Closed]], we have that $I_S$ is [[Definition:Closed Mapping|closed]].
We also have... | T2 Property is Preserved under Expansion | https://proofwiki.org/wiki/T2_Property_is_Preserved_under_Expansion | https://proofwiki.org/wiki/T2_Property_is_Preserved_under_Expansion | [
"Hausdorff Spaces",
"Separation Properties Preserved under Expansion"
] | [
"Definition:T2 Space"
] | [
"Definition:T2 Space",
"Definition:Expansion of Topology",
"Identity Mapping to Expansion is Closed",
"Definition:Closed Mapping",
"Identity Mapping is Bijection",
"T2 Property is Preserved under Closed Bijection",
"Definition:T2 Space",
"Definition:T2 Space"
] |
proofwiki-23259 | T2.5 Property is Preserved under Expansion | If $T_A$ is a $T_{2 \frac 1 2}$ space, then so is $T_B$. | Let $T_A = \struct {S, \tau_1}$ be a $T_{2 \frac 1 2}$ space.
Let $T_B = \struct {S, \tau_2}$ be such that $\tau_2$ is an expansion of $\tau_1$.
From Identity Mapping to Expansion is Closed, we have that $I_S$ is closed.
We also have Identity Mapping is Bijection.
Hence we can directly apply:
:$T_{2 \frac 1 2}$ Propert... | If $T_A$ is a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]], then so is $T_B$. | Let $T_A = \struct {S, \tau_1}$ be a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]].
Let $T_B = \struct {S, \tau_2}$ be such that $\tau_2$ is an [[Definition:Expansion of Topology|expansion]] of $\tau_1$.
From [[Identity Mapping to Expansion is Closed]], we have that $I_S$ is [[Definition:Closed Mapping|closed]].... | T2.5 Property is Preserved under Expansion | https://proofwiki.org/wiki/T2.5_Property_is_Preserved_under_Expansion | https://proofwiki.org/wiki/T2.5_Property_is_Preserved_under_Expansion | [
"T2.5 Spaces",
"Separation Properties Preserved under Expansion"
] | [
"Definition:T2.5 Space"
] | [
"Definition:T2.5 Space",
"Definition:Expansion of Topology",
"Identity Mapping to Expansion is Closed",
"Definition:Closed Mapping",
"Identity Mapping is Bijection",
"T2.5 Property is Preserved under Closed Bijection",
"Definition:T2.5 Space",
"Definition:T2.5 Space"
] |
proofwiki-23260 | T3 Property is Not Preserved under Expansion | If $T_A$ is a $T_3$ space, then it is not necessarily the case that so is $T_B$. | ;Proof by Counterexample
Let $\struct {\R, \tau_1}$ be the set of real numbers under the usual (Euclidean) topology.
Let $\struct {\R, \tau_2}$ be the indiscrete rational extension of $\struct {\R, \tau_1}$.
By definition, $\struct {\R, \tau_2}$ is an expansion of $\struct {\R, \tau_1}$.
From Metric Space is $T_3$:
:$\... | If $T_A$ is a [[Definition:T3 Space|$T_3$ space]], then it is not necessarily the case that so is $T_B$. | ;[[Proof by Counterexample]]
Let $\struct {\R, \tau_1}$ be the [[Definition:Real Number|set of real numbers]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]].
Let $\struct {\R, \tau_2}$ be the [[Definition:Indiscrete Rational Extension of Reals|indiscrete rational extension]... | T3 Property is Not Preserved under Expansion | https://proofwiki.org/wiki/T3_Property_is_Not_Preserved_under_Expansion | https://proofwiki.org/wiki/T3_Property_is_Not_Preserved_under_Expansion | [
"T3 Spaces",
"Separation Properties Not Preserved under Expansion"
] | [
"Definition:T3 Space"
] | [
"Proof by Counterexample",
"Definition:Real Number",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Indiscrete Extension of Reals/Rational",
"Definition:Expansion of Topology",
"Metric Space is T3",
"Definition:T3 Space",
"Indiscrete Extension of Reals is not T3",
"Cat... |
proofwiki-23261 | Harmonic Continued Fraction for Pi | :$\dfrac 2 {\pi - 2} = \dfrac 1 1 + \cfrac 1 {\dfrac 1 2 + \cfrac 1 {\dfrac 1 3 + \cfrac 1 {\dfrac 1 4 + \cfrac 1 {\dfrac 1 5 + \cfrac 1 {\dfrac 1 6 + \cfrac \ddots \ddots} } } } }$ | Let $I_n = \ds \int_0^{\frac \pi 4} \dfrac {\paren {\cos 2 x}^n} {\cos^2 x} \rd x$.
With a view to expressing the primitive in the form:
{{begin-eqn}}
{{eqn | l = \int u \frac {\d v} {\d x} \rd x
| r = u v - \int v \frac {\d u} {\d x} \rd x
| c = Integration by Parts
}}
{{end-eqn}}
let:
{{begin-eqn}}
{{eqn ... | :$\dfrac 2 {\pi - 2} = \dfrac 1 1 + \cfrac 1 {\dfrac 1 2 + \cfrac 1 {\dfrac 1 3 + \cfrac 1 {\dfrac 1 4 + \cfrac 1 {\dfrac 1 5 + \cfrac 1 {\dfrac 1 6 + \cfrac \ddots \ddots} } } } }$ | Let $I_n = \ds \int_0^{\frac \pi 4} \dfrac {\paren {\cos 2 x}^n} {\cos^2 x} \rd x$.
With a view to expressing the [[Definition:Primitive|primitive]] in the form:
{{begin-eqn}}
{{eqn | l = \int u \frac {\d v} {\d x} \rd x
| r = u v - \int v \frac {\d u} {\d x} \rd x
| c = [[Integration by Parts]]
}}
{{end... | Harmonic Continued Fraction for Pi | https://proofwiki.org/wiki/Harmonic_Continued_Fraction_for_Pi | https://proofwiki.org/wiki/Harmonic_Continued_Fraction_for_Pi | [
"Examples of Continued Fractions",
"Continued Fractions",
"Formulas for Pi",
"Harmonic Continued Fraction for Pi",
"Pi"
] | [] | [
"Definition:Primitive",
"Integration by Parts",
"Derivative of Composite Function",
"Derivative of Cosine Function",
"Power Rule for Derivatives",
"Primitive of Square of Secant Function",
"Integration by Parts",
"Tangent is Sine divided by Cosine",
"Definition:Fraction/Numerator",
"Definition:Fra... |
proofwiki-23262 | Metric Space is T1 | A metric space $M = \struct {A, d}$ is a $T_1$ space. | From Metric Space is $T_2$, $M$ is a $T_1$ space.
From $T_2$ Space is $T_1$ it follows that $M$ is a $T_1$ space.
{{qed}}
Category:Metric Space fulfils all Separation Axioms
Category:Examples of T1 Spaces
8f1b54yf4ivjqtdvmirz6w81bhf83me | A [[Definition:Metric Space|metric space]] $M = \struct {A, d}$ is a [[Definition:T1 Space|$T_1$ space]]. | From [[Metric Space is T2|Metric Space is $T_2$]], $M$ is a [[Definition:T1 Space|$T_1$ space]].
From [[T2 Space is T1|$T_2$ Space is $T_1$]] it follows that $M$ is a [[Definition:T1 Space|$T_1$ space]].
{{qed}}
[[Category:Metric Space fulfils all Separation Axioms]]
[[Category:Examples of T1 Spaces]]
8f1b54yf4ivjqtd... | Metric Space is T1 | https://proofwiki.org/wiki/Metric_Space_is_T1 | https://proofwiki.org/wiki/Metric_Space_is_T1 | [
"Metric Space fulfils all Separation Axioms",
"Examples of T1 Spaces"
] | [
"Definition:Metric Space",
"Definition:T1 Space"
] | [
"Metric Space is T2",
"Definition:T1 Space",
"T2 Space is T1",
"Definition:T1 Space",
"Category:Metric Space fulfils all Separation Axioms",
"Category:Examples of T1 Spaces"
] |
proofwiki-23263 | Metric Space is T0 | A metric space $M = \struct {A, d}$ is a $T_0$ space. | From Metric Space is $T_1$, $M$ is a $T_1$ space.
From $T_1$ Space is $T_0$ it follows that $M$ is a $T_0$ space.
{{qed}}
Category:Metric Space fulfils all Separation Axioms
Category:Examples of T0 Spaces
nckw1120qksd27ur9etfquhok49r3te | A [[Definition:Metric Space|metric space]] $M = \struct {A, d}$ is a [[Definition:T0 Space|$T_0$ space]]. | From [[Metric Space is T1|Metric Space is $T_1$]], $M$ is a [[Definition:T1 Space|$T_1$ space]].
From [[T1 Space is T0|$T_1$ Space is $T_0$]] it follows that $M$ is a [[Definition:T0 Space|$T_0$ space]].
{{qed}}
[[Category:Metric Space fulfils all Separation Axioms]]
[[Category:Examples of T0 Spaces]]
nckw1120qksd27u... | Metric Space is T0 | https://proofwiki.org/wiki/Metric_Space_is_T0 | https://proofwiki.org/wiki/Metric_Space_is_T0 | [
"Metric Space fulfils all Separation Axioms",
"Examples of T0 Spaces"
] | [
"Definition:Metric Space",
"Definition:T0 Space"
] | [
"Metric Space is T1",
"Definition:T1 Space",
"T1 Space is T0",
"Definition:T0 Space",
"Category:Metric Space fulfils all Separation Axioms",
"Category:Examples of T0 Spaces"
] |
proofwiki-23264 | Metric Space is Normal | A metric space $M = \struct {A, d}$ is a normal space. | {{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
We have that:
:a Metric Space is $T_4$
:a Metric Space is $T_1$.
Hence the result.
{{qed}}
Category:Metric Space fulfils all Separation Axioms
Category:Examples of Normal Spaces
9twedg5d85es49rml0o0qkuj9ezzz39 | A [[Definition:Metric Space|metric space]] $M = \struct {A, d}$ is a [[Definition:Normal Space|normal space]]. | {{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
We have that:
:a [[Metric Space is T4|Metric Space is $T_4$]]
:a [[Metric Space is T1|Metric Space is $T_1$]].
Hence the result.
{{qed}}
[[Category:Metric Space fulfils all Separation Axioms]]
[[Category:Examples of Normal Space... | Metric Space is Normal | https://proofwiki.org/wiki/Metric_Space_is_Normal | https://proofwiki.org/wiki/Metric_Space_is_Normal | [
"Metric Space fulfils all Separation Axioms",
"Examples of Normal Spaces"
] | [
"Definition:Metric Space",
"Definition:Normal Space"
] | [
"Metric Space is T4",
"Metric Space is T1",
"Category:Metric Space fulfils all Separation Axioms",
"Category:Examples of Normal Spaces"
] |
proofwiki-23265 | Metric Space is Completely Regular | A metric space $M = \struct {A, d}$ is a completely regular space. | We have:
* Metric Space is Normal
* Normal Space is Completely Regular
Hence the result.
{{qed}}
Category:Metric Space fulfils all Separation Axioms
Category:Examples of Completely Regular Spaces
amdzzwkx14h4m77l8xn7qfrvg6u9rku | A [[Definition:Metric Space|metric space]] $M = \struct {A, d}$ is a [[Definition:Completely Regular Space|completely regular space]]. | We have:
* [[Metric Space is Normal]]
* [[Normal Space is Completely Regular]]
Hence the result.
{{qed}}
[[Category:Metric Space fulfils all Separation Axioms]]
[[Category:Examples of Completely Regular Spaces]]
amdzzwkx14h4m77l8xn7qfrvg6u9rku | Metric Space is Completely Regular | https://proofwiki.org/wiki/Metric_Space_is_Completely_Regular | https://proofwiki.org/wiki/Metric_Space_is_Completely_Regular | [
"Metric Space fulfils all Separation Axioms",
"Examples of Completely Regular Spaces"
] | [
"Definition:Metric Space",
"Definition:Completely Regular Space"
] | [
"Metric Space is Normal",
"Normal Space is Completely Regular",
"Category:Metric Space fulfils all Separation Axioms",
"Category:Examples of Completely Regular Spaces"
] |
proofwiki-23266 | Metric Space is T3.5 | A metric space $M = \struct {A, d}$ is a $T_{3 \frac 1 2}$ space. | {{Recall|Completely Regular Space}}
{{:Definition:Completely Regular Space}}
We have:
:Metric Space is Completely Regular
Hence the result.
{{qed}}
Category:Metric Space fulfils all Separation Axioms
Category:Examples of T3.5 Spaces
rlaqi47dyzetkdkqesypcbo4yerbtuf | A [[Definition:Metric Space|metric space]] $M = \struct {A, d}$ is a [[Definition:T3.5 Space|$T_{3 \frac 1 2}$ space]]. | {{Recall|Completely Regular Space}}
{{:Definition:Completely Regular Space}}
We have:
:[[Metric Space is Completely Regular]]
Hence the result.
{{qed}}
[[Category:Metric Space fulfils all Separation Axioms]]
[[Category:Examples of T3.5 Spaces]]
rlaqi47dyzetkdkqesypcbo4yerbtuf | Metric Space is T3.5 | https://proofwiki.org/wiki/Metric_Space_is_T3.5 | https://proofwiki.org/wiki/Metric_Space_is_T3.5 | [
"Metric Space fulfils all Separation Axioms",
"Examples of T3.5 Spaces"
] | [
"Definition:Metric Space",
"Definition:T3.5 Space"
] | [
"Metric Space is Completely Regular",
"Category:Metric Space fulfils all Separation Axioms",
"Category:Examples of T3.5 Spaces"
] |
proofwiki-23267 | Metric Space is Urysohn | A metric space $M = \struct {A, d}$ is an Urysohn space. | We have:
:Metric Space is Completely Regular
:Completely Regular Space is Urysohn.
Hence the result.
{{qed}}
Category:Metric Space fulfils all Separation Axioms
Category:Examples of Urysohn Spaces
ikofrwbmmmto628fmz8r65wrwfr0h1d | A [[Definition:Metric Space|metric space]] $M = \struct {A, d}$ is an [[Definition:Urysohn Space|Urysohn space]]. | We have:
:[[Metric Space is Completely Regular]]
:[[Completely Regular Space is Urysohn]].
Hence the result.
{{qed}}
[[Category:Metric Space fulfils all Separation Axioms]]
[[Category:Examples of Urysohn Spaces]]
ikofrwbmmmto628fmz8r65wrwfr0h1d | Metric Space is Urysohn | https://proofwiki.org/wiki/Metric_Space_is_Urysohn | https://proofwiki.org/wiki/Metric_Space_is_Urysohn | [
"Metric Space fulfils all Separation Axioms",
"Examples of Urysohn Spaces"
] | [
"Definition:Metric Space",
"Definition:Urysohn Space"
] | [
"Metric Space is Completely Regular",
"Completely Regular Space is Urysohn",
"Category:Metric Space fulfils all Separation Axioms",
"Category:Examples of Urysohn Spaces"
] |
proofwiki-23268 | Metric Space is Regular | A metric space $M = \struct {A, d}$ is a regular space. | We have:
:Metric Space is Completely Regular
:Completely Regular Space is Regular.
Hence the result.
{{qed}} | A [[Definition:Metric Space|metric space]] $M = \struct {A, d}$ is a [[Definition:Regular Space|regular space]]. | We have:
:[[Metric Space is Completely Regular]]
:[[Completely Regular Space is Regular]].
Hence the result.
{{qed}} | Metric Space is Regular/Proof 1 | https://proofwiki.org/wiki/Metric_Space_is_Regular | https://proofwiki.org/wiki/Metric_Space_is_Regular/Proof_1 | [
"Metric Space is Regular",
"Metric Space fulfils all Separation Axioms",
"Examples of Regular Spaces"
] | [
"Definition:Metric Space",
"Definition:Regular Space"
] | [
"Metric Space is Completely Regular",
"Completely Regular Space is Regular"
] |
proofwiki-23269 | Metric Space is Regular | A metric space $M = \struct {A, d}$ is a regular space. | We have:
:Metric Space is Fully Normal
:Fully Normal Space is Normal
:Normal Space is Regular.
Hence the result.
{{qed}} | A [[Definition:Metric Space|metric space]] $M = \struct {A, d}$ is a [[Definition:Regular Space|regular space]]. | We have:
:[[Metric Space is Fully Normal]]
:[[Fully Normal Space is Normal]]
:[[Normal Space is Regular]].
Hence the result.
{{qed}} | Metric Space is Regular/Proof 2 | https://proofwiki.org/wiki/Metric_Space_is_Regular | https://proofwiki.org/wiki/Metric_Space_is_Regular/Proof_2 | [
"Metric Space is Regular",
"Metric Space fulfils all Separation Axioms",
"Examples of Regular Spaces"
] | [
"Definition:Metric Space",
"Definition:Regular Space"
] | [
"Metric Space is Fully Normal",
"Fully Normal Space is Normal",
"Normal Space is Regular"
] |
proofwiki-23270 | Completely Regular Space is T1 | Let $\struct {S, \tau}$ be a completely regular space.
Then $\struct {S, \tau}$ is also a $T_1$ space. | We have:
:Completely Regular Space is $T_2$
:$T_2$ Space is $T_1$
Hence the result.
{{qed}} | Let $\struct {S, \tau}$ be a [[Definition:Completely Regular Space|completely regular space]].
Then $\struct {S, \tau}$ is also a [[Definition:T1 Space|$T_1$ space]]. | We have:
:[[Completely Regular Space is T2|Completely Regular Space is $T_2$]]
:[[T2 Space is T1|$T_2$ Space is $T_1$]]
Hence the result.
{{qed}} | Completely Regular Space is T1 | https://proofwiki.org/wiki/Completely_Regular_Space_is_T1 | https://proofwiki.org/wiki/Completely_Regular_Space_is_T1 | [
"Completely Regular Spaces",
"T1 Spaces"
] | [
"Definition:Completely Regular Space",
"Definition:T1 Space"
] | [
"Completely Regular Space is T2",
"T2 Space is T1"
] |
proofwiki-23271 | Completely Regular Space is T2 | Let $\struct {S, \tau}$ be a completely regular space.
Then $\struct {S, \tau}$ is also a $T_2$ (Hausdorff) space. | {{Recall|Regular Space|regular space}}
{{:Definition:Regular Space/Definition 3}}
We have:
:Completely Regular Space is Regular
Hence the result.
{{qed}} | Let $\struct {S, \tau}$ be a [[Definition:Completely Regular Space|completely regular space]].
Then $\struct {S, \tau}$ is also a [[Definition:T2 Space|$T_2$ (Hausdorff) space]]. | {{Recall|Regular Space|regular space}}
{{:Definition:Regular Space/Definition 3}}
We have:
:[[Completely Regular Space is Regular]]
Hence the result.
{{qed}} | Completely Regular Space is T2 | https://proofwiki.org/wiki/Completely_Regular_Space_is_T2 | https://proofwiki.org/wiki/Completely_Regular_Space_is_T2 | [
"Completely Regular Spaces",
"Hausdorff Spaces"
] | [
"Definition:Completely Regular Space",
"Definition:T2 Space"
] | [
"Completely Regular Space is Regular"
] |
proofwiki-23272 | Metric Space is T3 | A metric space $M = \struct {A, d}$ is a $T_3$ space. | {{Recall|Regular Space|index = 1}}
{{:Definition:Regular Space/Definition 1}}
We have:
:Metric Space is Regular Space
Hence the result.
{{qed}}
Category:Metric Space fulfils all Separation Axioms
Category:Examples of T3 Spaces
o73eog6uxvibvpuwms8bvxr43fo15h0 | A [[Definition:Metric Space|metric space]] $M = \struct {A, d}$ is a [[Definition:T3 Space|$T_3$ space]]. | {{Recall|Regular Space|index = 1}}
{{:Definition:Regular Space/Definition 1}}
We have:
:[[Metric Space is Regular Space]]
Hence the result.
{{qed}}
[[Category:Metric Space fulfils all Separation Axioms]]
[[Category:Examples of T3 Spaces]]
o73eog6uxvibvpuwms8bvxr43fo15h0 | Metric Space is T3 | https://proofwiki.org/wiki/Metric_Space_is_T3 | https://proofwiki.org/wiki/Metric_Space_is_T3 | [
"Metric Space fulfils all Separation Axioms",
"Examples of T3 Spaces"
] | [
"Definition:Metric Space",
"Definition:T3 Space"
] | [
"Metric Space is Regular",
"Category:Metric Space fulfils all Separation Axioms",
"Category:Examples of T3 Spaces"
] |
proofwiki-23273 | Metric Space is T2.5 | A metric space $M = \struct {A, d}$ is a $T_{2 \frac 1 2}$ space. | We have:
:Metric Space is Regular
:Regular Space is $T_{2 \frac 1 2}$ Space
Hence the result.
{{qed}}
Category:Metric Space fulfils all Separation Axioms
Category:Examples of T2.5 Spaces
si4kug551qywib65obvw4um48t4vter | A [[Definition:Metric Space|metric space]] $M = \struct {A, d}$ is a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]]. | We have:
:[[Metric Space is Regular]]
:[[Regular Space is T2.5|Regular Space is $T_{2 \frac 1 2}$ Space]]
Hence the result.
{{qed}}
[[Category:Metric Space fulfils all Separation Axioms]]
[[Category:Examples of T2.5 Spaces]]
si4kug551qywib65obvw4um48t4vter | Metric Space is T2.5 | https://proofwiki.org/wiki/Metric_Space_is_T2.5 | https://proofwiki.org/wiki/Metric_Space_is_T2.5 | [
"Metric Space fulfils all Separation Axioms",
"Examples of T2.5 Spaces"
] | [
"Definition:Metric Space",
"Definition:T2.5 Space"
] | [
"Metric Space is Regular",
"Regular Space is T2.5",
"Category:Metric Space fulfils all Separation Axioms",
"Category:Examples of T2.5 Spaces"
] |
proofwiki-23274 | Metric Space is Semiregular | A metric space $M = \struct {A, d}$ is a semiregular space. | We have:
:Metric Space is Regular
:Regular Space is Semiregular.
Hence the result.
{{qed}}
Category:Metric Space fulfils all Separation Axioms
Category:Examples of Semiregular Spaces
d4dkcb119jsvxld73mt69idx08lc16g | A [[Definition:Metric Space|metric space]] $M = \struct {A, d}$ is a [[Definition:Semiregular Space|semiregular space]]. | We have:
:[[Metric Space is Regular]]
:[[Regular Space is Semiregular]].
Hence the result.
{{qed}}
[[Category:Metric Space fulfils all Separation Axioms]]
[[Category:Examples of Semiregular Spaces]]
d4dkcb119jsvxld73mt69idx08lc16g | Metric Space is Semiregular | https://proofwiki.org/wiki/Metric_Space_is_Semiregular | https://proofwiki.org/wiki/Metric_Space_is_Semiregular | [
"Metric Space fulfils all Separation Axioms",
"Examples of Semiregular Spaces"
] | [
"Definition:Metric Space",
"Definition:Semiregular Space"
] | [
"Metric Space is Regular",
"Regular Space is Semiregular",
"Category:Metric Space fulfils all Separation Axioms",
"Category:Examples of Semiregular Spaces"
] |
proofwiki-23275 | T4 Property is Not Preserved under Expansion | If $T_A$ is a $T_4$ space, then it is not necessarily the case that so is $T_B$. | ;Proof by Counterexample
Let $\struct {\R, \tau_1}$ be the set of real numbers under the usual (Euclidean) topology.
Let $\struct {\R, \tau_2}$ be the indiscrete rational extension of $\struct {\R, \tau_1}$.
By definition, $\struct {\R, \tau_2}$ is an expansion of $\struct {\R, \tau_1}$.
From Metric Space is $T_4$:
:$\... | If $T_A$ is a [[Definition:T4 Space|$T_4$ space]], then it is not necessarily the case that so is $T_B$. | ;[[Proof by Counterexample]]
Let $\struct {\R, \tau_1}$ be the [[Definition:Real Number|set of real numbers]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]].
Let $\struct {\R, \tau_2}$ be the [[Definition:Indiscrete Rational Extension of Reals|indiscrete rational extension]... | T4 Property is Not Preserved under Expansion | https://proofwiki.org/wiki/T4_Property_is_Not_Preserved_under_Expansion | https://proofwiki.org/wiki/T4_Property_is_Not_Preserved_under_Expansion | [
"T4 Spaces",
"Separation Properties Not Preserved under Expansion"
] | [
"Definition:T4 Space"
] | [
"Proof by Counterexample",
"Definition:Real Number",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Indiscrete Extension of Reals/Rational",
"Definition:Expansion of Topology",
"Metric Space is T4",
"Definition:T4 Space",
"Indiscrete Extension of Reals is not T4",
"Cat... |
proofwiki-23276 | T5 Property is Not Preserved under Expansion | If $T_A$ is a $T_5$ space, then it is not necessarily the case that so is $T_B$. | ;Proof by Counterexample
Let $\struct {\R, \tau_1}$ be the set of real numbers under the usual (Euclidean) topology.
Let $\struct {\R, \tau_2}$ be the indiscrete rational extension of $\struct {\R, \tau_1}$.
By definition, $\struct {\R, \tau_2}$ is an expansion of $\struct {\R, \tau_1}$.
From Metric Space is $T_5$:
:$\... | If $T_A$ is a [[Definition:T5 Space|$T_5$ space]], then it is not necessarily the case that so is $T_B$. | ;[[Proof by Counterexample]]
Let $\struct {\R, \tau_1}$ be the [[Definition:Real Number|set of real numbers]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]].
Let $\struct {\R, \tau_2}$ be the [[Definition:Indiscrete Rational Extension of Reals|indiscrete rational extension]... | T5 Property is Not Preserved under Expansion | https://proofwiki.org/wiki/T5_Property_is_Not_Preserved_under_Expansion | https://proofwiki.org/wiki/T5_Property_is_Not_Preserved_under_Expansion | [
"T5 Spaces",
"Separation Properties Not Preserved under Expansion"
] | [
"Definition:T5 Space"
] | [
"Proof by Counterexample",
"Definition:Real Number",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Indiscrete Extension of Reals/Rational",
"Definition:Expansion of Topology",
"Metric Space is T5",
"Definition:T5 Space",
"Indiscrete Extension of Reals is not T5",
"Cat... |
proofwiki-23277 | Regular Property is Not Preserved under Expansion | If $T_A$ is a regular space, then it is not necessarily the case that so is $T_B$. | Let $T_A$ be a regular space.
{{Recall|Regular Space|index = 1}}
{{:Definition:Regular Space/Definition 1}}
Thus $T_A$ is a $T_3$ space.
From $T_0$ Property is Preserved under Expansion we have that $T_B$ is a $T_0$ space.
But from $T_3$ Property is Not Preserved under Expansion, it is not necessarily the case that $T_... | If $T_A$ is a [[Definition:Regular Space|regular space]], then it is not necessarily the case that so is $T_B$. | Let $T_A$ be a [[Definition:Regular Space|regular space]].
{{Recall|Regular Space|index = 1}}
{{:Definition:Regular Space/Definition 1}}
Thus $T_A$ is a [[Definition:T3 Space|$T_3$ space]].
From [[T0 Property is Preserved under Expansion|$T_0$ Property is Preserved under Expansion]] we have that $T_B$ is a [[Defini... | Regular Property is Not Preserved under Expansion | https://proofwiki.org/wiki/Regular_Property_is_Not_Preserved_under_Expansion | https://proofwiki.org/wiki/Regular_Property_is_Not_Preserved_under_Expansion | [
"Regular Spaces",
"Separation Properties Not Preserved under Expansion"
] | [
"Definition:Regular Space"
] | [
"Definition:Regular Space",
"Definition:T3 Space",
"T0 Property is Preserved under Expansion",
"Definition:T0 Space",
"T3 Property is Not Preserved under Expansion",
"Definition:T3 Space",
"Definition:Regular Space",
"Category:Regular Spaces",
"Category:Separation Properties Not Preserved under Expa... |
proofwiki-23278 | T3.5 Property is Not Preserved under Expansion | If $T_A$ is a $T_{3 \frac 1 2}$ space, then it is not necessarily the case that so is $T_B$. | Let $T_A$ be a $T_{3 \frac 1 2}$ space.
From $T_{3 \frac 1 2}$ Space is $T_3$, it follows that $T_A$ is a $T_3$ space.
But from $T_3$ Property is Not Preserved under Expansion, it is not necessarily the case that $T_B$ is also a $T_3$ space.
But again, we have:
:$T_{3 \frac 1 2}$ Space is $T_3$
Hence, by the Rule of Tr... | If $T_A$ is a [[Definition:T3.5 Space|$T_{3 \frac 1 2}$ space]], then it is not necessarily the case that so is $T_B$. | Let $T_A$ be a [[Definition:T3.5 Space|$T_{3 \frac 1 2}$ space]].
From [[T3.5 Space is T3|$T_{3 \frac 1 2}$ Space is $T_3$]], it follows that $T_A$ is a [[Definition:T3 Space|$T_3$ space]].
But from [[T3 Property is Not Preserved under Expansion|$T_3$ Property is Not Preserved under Expansion]], it is not necessarily... | T3.5 Property is Not Preserved under Expansion | https://proofwiki.org/wiki/T3.5_Property_is_Not_Preserved_under_Expansion | https://proofwiki.org/wiki/T3.5_Property_is_Not_Preserved_under_Expansion | [
"T3.5 Spaces",
"Separation Properties Not Preserved under Expansion"
] | [
"Definition:T3.5 Space"
] | [
"Definition:T3.5 Space",
"T3.5 Space is T3",
"Definition:T3 Space",
"T3 Property is Not Preserved under Expansion",
"Definition:T3 Space",
"T3.5 Space is T3",
"Rule of Transposition",
"Definition:T3.5 Space",
"Category:T3.5 Spaces",
"Category:Separation Properties Not Preserved under Expansion"
] |
proofwiki-23279 | Completely Regular Property is Not Preserved under Expansion | If $T_A$ is a completely regular space, then it is not necessarily the case that so is $T_B$. | Let $T_A$ be a completely regular space.
From Completely Regular Space is Regular, it follows that $T_A$ is a regular space.
But from Regular Property is Not Preserved under Expansion, it is not necessarily the case that $T_B$ is also a regular space.
But again, we have:
:Completely Regular Space is Regular
Hence, by t... | If $T_A$ is a [[Definition:Completely Regular Space|completely regular space]], then it is not necessarily the case that so is $T_B$. | Let $T_A$ be a [[Definition:Completely Regular Space|completely regular space]].
From [[Completely Regular Space is Regular]], it follows that $T_A$ is a [[Definition:Regular Space|regular space]].
But from [[Regular Property is Not Preserved under Expansion]], it is not necessarily the case that $T_B$ is also a [[De... | Completely Regular Property is Not Preserved under Expansion | https://proofwiki.org/wiki/Completely_Regular_Property_is_Not_Preserved_under_Expansion | https://proofwiki.org/wiki/Completely_Regular_Property_is_Not_Preserved_under_Expansion | [
"Completely Regular Spaces",
"Separation Properties Not Preserved under Expansion"
] | [
"Definition:Completely Regular Space"
] | [
"Definition:Completely Regular Space",
"Completely Regular Space is Regular",
"Definition:Regular Space",
"Regular Property is Not Preserved under Expansion",
"Definition:Regular Space",
"Completely Regular Space is Regular",
"Rule of Transposition",
"Definition:Completely Regular Space",
"Category:... |
proofwiki-23280 | Normal Property is Not Preserved under Expansion | If $T_A$ is a normal space, then it is not necessarily the case that so is $T_B$. | Let $T_A$ be a normal space.
{{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
Thus $T_A$ is a $T_4$ space.
From $T_1$ Property is Preserved under Expansion we have that $T_B$ is a $T_1$ space.
From $T_4$ Property is Not Preserved under Expansion, it is not necessarily the case th... | If $T_A$ is a [[Definition:Normal Space|normal space]], then it is not necessarily the case that so is $T_B$. | Let $T_A$ be a [[Definition:Normal Space|normal space]].
{{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
Thus $T_A$ is a [[Definition:T4 Space|$T_4$ space]].
From [[T1 Property is Preserved under Expansion|$T_1$ Property is Preserved under Expansion]] we have that $T_B$ is a... | Normal Property is Not Preserved under Expansion | https://proofwiki.org/wiki/Normal_Property_is_Not_Preserved_under_Expansion | https://proofwiki.org/wiki/Normal_Property_is_Not_Preserved_under_Expansion | [
"Normal Spaces",
"Separation Properties Not Preserved under Expansion"
] | [
"Definition:Normal Space"
] | [
"Definition:Normal Space",
"Definition:T4 Space",
"T1 Property is Preserved under Expansion",
"Definition:T1 Space",
"T4 Property is Not Preserved under Expansion",
"Definition:T4 Space",
"Definition:Normal Space",
"Category:Normal Spaces",
"Category:Separation Properties Not Preserved under Expansi... |
proofwiki-23281 | Completely Normal Property is Not Preserved under Expansion | If $T_A$ is a completely normal space, then it is not necessarily the case that so is $T_B$. | Let $T_A$ be a completely normal space.
{{Recall|Completely Normal Space|completely normal space|index = 1}}
{{:Definition:Completely Normal Space/Definition 1}}
Thus $T_A$ is a $T_5$ space.
From $T_1$ Property is Preserved under Expansion we have that $T_B$ is a $T_1$ space.
But from $T_5$ Property is Not Preserved un... | If $T_A$ is a [[Definition:Completely Normal Space|completely normal space]], then it is not necessarily the case that so is $T_B$. | Let $T_A$ be a [[Definition:Completely Normal Space|completely normal space]].
{{Recall|Completely Normal Space|completely normal space|index = 1}}
{{:Definition:Completely Normal Space/Definition 1}}
Thus $T_A$ is a [[Definition:T5 Space|$T_5$ space]].
From [[T1 Property is Preserved under Expansion|$T_1$ Property... | Completely Normal Property is Not Preserved under Expansion | https://proofwiki.org/wiki/Completely_Normal_Property_is_Not_Preserved_under_Expansion | https://proofwiki.org/wiki/Completely_Normal_Property_is_Not_Preserved_under_Expansion | [
"Completely Normal Spaces",
"Separation Properties Not Preserved under Expansion"
] | [
"Definition:Completely Normal Space"
] | [
"Definition:Completely Normal Space",
"Definition:T5 Space",
"T1 Property is Preserved under Expansion",
"Definition:T1 Space",
"T5 Property is Not Preserved under Expansion",
"Definition:T5 Space",
"Definition:Completely Normal Space",
"Category:Completely Normal Spaces",
"Category:Separation Prope... |
proofwiki-23282 | Indiscrete Extension of Reals is not Regular | Let $T$ be an indiscrete extension of $\R$.
Then $T$ is not a regular space. | {{Recall|Regular Space|index = 1}}
{{:Definition:Regular Space/Definition 1}}
From Indiscrete Extension of Reals is $T_0$, we have that $T$ is a $T_0$ space.
From Indiscrete Extension of Reals is not $T_3$, we have that $T$ is not a $T_3$ space.
Hence the result.
{{qed}}
Category:Indiscrete Extensions of Reals
Category... | Let $T$ be an [[Definition:Indiscrete Extension of Reals|indiscrete extension of $\R$]].
Then $T$ is not a [[Definition:Regular Space|regular space]]. | {{Recall|Regular Space|index = 1}}
{{:Definition:Regular Space/Definition 1}}
From [[Indiscrete Extension of Reals is T0|Indiscrete Extension of Reals is $T_0$]], we have that $T$ is a [[Definition:T0 Space|$T_0$ space]].
From [[Indiscrete Extension of Reals is not T3|Indiscrete Extension of Reals is not $T_3$]], we ... | Indiscrete Extension of Reals is not Regular | https://proofwiki.org/wiki/Indiscrete_Extension_of_Reals_is_not_Regular | https://proofwiki.org/wiki/Indiscrete_Extension_of_Reals_is_not_Regular | [
"Indiscrete Extensions of Reals",
"Examples of Regular Spaces"
] | [
"Definition:Indiscrete Extension of Reals",
"Definition:Regular Space"
] | [
"Indiscrete Extension of Reals is T0",
"Definition:T0 Space",
"Indiscrete Extension of Reals is not T3",
"Definition:T3 Space",
"Category:Indiscrete Extensions of Reals",
"Category:Examples of Regular Spaces"
] |
proofwiki-23283 | Indiscrete Extension of Reals is not Normal | Let $T$ be an indiscrete extension of $\R$.
Then $T$ is not a normal space. | {{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
From Indiscrete Extension of Reals is $T_1$, we have that $T$ is a $T_1$ space.
From Indiscrete Extension of Reals is not $T_4$, we have that $T$ is not a $T_4$ space.
Hence the result.
{{qed}}
Category:Indiscrete Extensions of Rea... | Let $T$ be an [[Definition:Indiscrete Extension of Reals|indiscrete extension of $\R$]].
Then $T$ is not a [[Definition:Normal Space|normal space]]. | {{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
From [[Indiscrete Extension of Reals is T1|Indiscrete Extension of Reals is $T_1$]], we have that $T$ is a [[Definition:T1 Space|$T_1$ space]].
From [[Indiscrete Extension of Reals is not T4|Indiscrete Extension of Reals is not $... | Indiscrete Extension of Reals is not Normal | https://proofwiki.org/wiki/Indiscrete_Extension_of_Reals_is_not_Normal | https://proofwiki.org/wiki/Indiscrete_Extension_of_Reals_is_not_Normal | [
"Indiscrete Extensions of Reals",
"Examples of Normal Spaces"
] | [
"Definition:Indiscrete Extension of Reals",
"Definition:Normal Space"
] | [
"Indiscrete Extension of Reals is T1",
"Definition:T1 Space",
"Indiscrete Extension of Reals is not T4",
"Definition:T4 Space",
"Category:Indiscrete Extensions of Reals",
"Category:Examples of Normal Spaces"
] |
proofwiki-23284 | Indiscrete Extension of Reals is not Completely Normal | Let $T$ be an indiscrete extension of $\R$.
Then $T$ is not a completely normal space. | {{Recall|Completely Normal Space|completely normal space|index = 1}}
{{:Definition:Completely Normal Space/Definition 1}}
From Indiscrete Extension of Reals is $T_1$, we have that $T$ is a $T_1$ space.
From Indiscrete Extension of Reals is not $T_5$, we have that $T$ is not a $T_5$ space.
Hence the result.
{{qed}}
Cate... | Let $T$ be an [[Definition:Indiscrete Extension of Reals|indiscrete extension of $\R$]].
Then $T$ is not a [[Definition:Completely Normal Space|completely normal space]]. | {{Recall|Completely Normal Space|completely normal space|index = 1}}
{{:Definition:Completely Normal Space/Definition 1}}
From [[Indiscrete Extension of Reals is T1|Indiscrete Extension of Reals is $T_1$]], we have that $T$ is a [[Definition:T1 Space|$T_1$ space]].
From [[Indiscrete Extension of Reals is not T5|Indis... | Indiscrete Extension of Reals is not Completely Normal | https://proofwiki.org/wiki/Indiscrete_Extension_of_Reals_is_not_Completely_Normal | https://proofwiki.org/wiki/Indiscrete_Extension_of_Reals_is_not_Completely_Normal | [
"Indiscrete Extensions of Reals",
"Examples of Completely Normal Spaces"
] | [
"Definition:Indiscrete Extension of Reals",
"Definition:Completely Normal Space"
] | [
"Indiscrete Extension of Reals is T1",
"Definition:T1 Space",
"Indiscrete Extension of Reals is not T5",
"Definition:T5 Space",
"Category:Indiscrete Extensions of Reals",
"Category:Examples of Completely Normal Spaces"
] |
proofwiki-23285 | Real Number Line with Euclidean Topology is T0 | :$\struct {\R, \tau_d}$ is a $T_0$ space | From Real Number Line is Metric Space we have that $\struct {\R, \tau_d}$ is an example of a metric space.
The result follows from Metric Space is $T_0$.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of T0 Spaces
4b7xcpig775xg62ukcx... | :$\struct {\R, \tau_d}$ is a [[Definition:T0 Space|$T_0$ space]] | From [[Real Number Line is Metric Space]] we have that $\struct {\R, \tau_d}$ is an example of a [[Definition:Metric Space|metric space]].
The result follows from [[Metric Space is T0|Metric Space is $T_0$]].
{{qed}}
[[Category:Real Number Line satisfies all Separation Axioms]]
[[Category:Real Number Line with Euclid... | Real Number Line with Euclidean Topology is T0 | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_T0 | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_T0 | [
"Real Number Line satisfies all Separation Axioms",
"Real Number Line with Euclidean Topology",
"Examples of T0 Spaces"
] | [
"Definition:T0 Space"
] | [
"Real Number Line is Metric Space",
"Definition:Metric Space",
"Metric Space is T0",
"Category:Real Number Line satisfies all Separation Axioms",
"Category:Real Number Line with Euclidean Topology",
"Category:Examples of T0 Spaces"
] |
proofwiki-23286 | Real Number Line with Euclidean Topology is T1 | :$\struct {\R, \tau_d}$ is a $T_1$ space | From Real Number Line is Metric Space we have that $\struct {\R, \tau_d}$ is an example of a metric space.
The result follows from Metric Space is $T_1$.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of T1 Spaces
s5psh1ltgr499skda9l... | :$\struct {\R, \tau_d}$ is a [[Definition:T1 Space|$T_1$ space]] | From [[Real Number Line is Metric Space]] we have that $\struct {\R, \tau_d}$ is an example of a [[Definition:Metric Space|metric space]].
The result follows from [[Metric Space is T1|Metric Space is $T_1$]].
{{qed}}
[[Category:Real Number Line satisfies all Separation Axioms]]
[[Category:Real Number Line with Euclid... | Real Number Line with Euclidean Topology is T1 | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_T1 | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_T1 | [
"Real Number Line satisfies all Separation Axioms",
"Real Number Line with Euclidean Topology",
"Examples of T1 Spaces"
] | [
"Definition:T1 Space"
] | [
"Real Number Line is Metric Space",
"Definition:Metric Space",
"Metric Space is T1",
"Category:Real Number Line satisfies all Separation Axioms",
"Category:Real Number Line with Euclidean Topology",
"Category:Examples of T1 Spaces"
] |
proofwiki-23287 | Real Number Line with Euclidean Topology is T2 | :$\struct {\R, \tau_d}$ is a $T_2$ (Hausdorff) space | From Real Number Line is Metric Space we have that $\struct {\R, \tau_d}$ is an example of a metric space.
The result follows from Metric Space is $T_2$.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of Hausdorff Spaces
i2s3u94unr54... | :$\struct {\R, \tau_d}$ is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]] | From [[Real Number Line is Metric Space]] we have that $\struct {\R, \tau_d}$ is an example of a [[Definition:Metric Space|metric space]].
The result follows from [[Metric Space is T2|Metric Space is $T_2$]].
{{qed}}
[[Category:Real Number Line satisfies all Separation Axioms]]
[[Category:Real Number Line with Euclid... | Real Number Line with Euclidean Topology is T2 | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_T2 | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_T2 | [
"Real Number Line satisfies all Separation Axioms",
"Real Number Line with Euclidean Topology",
"Examples of Hausdorff Spaces"
] | [
"Definition:T2 Space"
] | [
"Real Number Line is Metric Space",
"Definition:Metric Space",
"Metric Space is T2",
"Category:Real Number Line satisfies all Separation Axioms",
"Category:Real Number Line with Euclidean Topology",
"Category:Examples of Hausdorff Spaces"
] |
proofwiki-23288 | Real Number Line with Euclidean Topology is Semiregular | :$\struct {\R, \tau_d}$ is a semiregular space | From Real Number Line is Metric Space we have that $\struct {\R, \tau_d}$ is an example of a metric space.
The result follows from Metric Space is Semiregular.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of Semiregular Spaces
9e3z... | :$\struct {\R, \tau_d}$ is a [[Definition:Semiregular Space|semiregular space]] | From [[Real Number Line is Metric Space]] we have that $\struct {\R, \tau_d}$ is an example of a [[Definition:Metric Space|metric space]].
The result follows from [[Metric Space is Semiregular]].
{{qed}}
[[Category:Real Number Line satisfies all Separation Axioms]]
[[Category:Real Number Line with Euclidean Topology]... | Real Number Line with Euclidean Topology is Semiregular | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_Semiregular | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_Semiregular | [
"Real Number Line satisfies all Separation Axioms",
"Real Number Line with Euclidean Topology",
"Examples of Semiregular Spaces"
] | [
"Definition:Semiregular Space"
] | [
"Real Number Line is Metric Space",
"Definition:Metric Space",
"Metric Space is Semiregular",
"Category:Real Number Line satisfies all Separation Axioms",
"Category:Real Number Line with Euclidean Topology",
"Category:Examples of Semiregular Spaces"
] |
proofwiki-23289 | Real Number Line with Euclidean Topology is T2.5 | :$\struct {\R, \tau_d}$ is a $T_{2 \frac 1 2}$ space | From Real Number Line is Metric Space we have that $\struct {\R, \tau_d}$ is an example of a metric space.
The result follows from Metric Space is $T_{2 \frac 1 2}$.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of T2.5 Spaces
n6fkf... | :$\struct {\R, \tau_d}$ is a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]] | From [[Real Number Line is Metric Space]] we have that $\struct {\R, \tau_d}$ is an example of a [[Definition:Metric Space|metric space]].
The result follows from [[Metric Space is T2.5|Metric Space is $T_{2 \frac 1 2}$]].
{{qed}}
[[Category:Real Number Line satisfies all Separation Axioms]]
[[Category:Real Number Li... | Real Number Line with Euclidean Topology is T2.5 | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_T2.5 | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_T2.5 | [
"Real Number Line satisfies all Separation Axioms",
"Real Number Line with Euclidean Topology",
"Examples of T2.5 Spaces"
] | [
"Definition:T2.5 Space"
] | [
"Real Number Line is Metric Space",
"Definition:Metric Space",
"Metric Space is T2.5",
"Category:Real Number Line satisfies all Separation Axioms",
"Category:Real Number Line with Euclidean Topology",
"Category:Examples of T2.5 Spaces"
] |
proofwiki-23290 | Real Number Line with Euclidean Topology is T3.5 | :$\struct {\R, \tau_d}$ is a $T_{3 \frac 1 2}$ space | From Real Number Line is Metric Space we have that $\struct {\R, \tau_d}$ is an example of a metric space.
The result follows from Metric Space is $T_{3 \frac 1 2}$.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of T3.5 Spaces
1p626... | :$\struct {\R, \tau_d}$ is a [[Definition:T3.5 Space|$T_{3 \frac 1 2}$ space]] | From [[Real Number Line is Metric Space]] we have that $\struct {\R, \tau_d}$ is an example of a [[Definition:Metric Space|metric space]].
The result follows from [[Metric Space is T3.5|Metric Space is $T_{3 \frac 1 2}$]].
{{qed}}
[[Category:Real Number Line satisfies all Separation Axioms]]
[[Category:Real Number Li... | Real Number Line with Euclidean Topology is T3.5 | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_T3.5 | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_T3.5 | [
"Real Number Line satisfies all Separation Axioms",
"Real Number Line with Euclidean Topology",
"Examples of T3.5 Spaces"
] | [
"Definition:T3.5 Space"
] | [
"Real Number Line is Metric Space",
"Definition:Metric Space",
"Metric Space is T3.5",
"Category:Real Number Line satisfies all Separation Axioms",
"Category:Real Number Line with Euclidean Topology",
"Category:Examples of T3.5 Spaces"
] |
proofwiki-23291 | Real Number Line with Euclidean Topology is T3 | :$\struct {\R, \tau_d}$ is a $T_3$ space | From Real Number Line is Metric Space we have that $\struct {\R, \tau_d}$ is an example of a metric space.
The result follows from Metric Space is $T_3$.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of T3 Spaces
b94laj2v76sw03q9co1... | :$\struct {\R, \tau_d}$ is a [[Definition:T3 Space|$T_3$ space]] | From [[Real Number Line is Metric Space]] we have that $\struct {\R, \tau_d}$ is an example of a [[Definition:Metric Space|metric space]].
The result follows from [[Metric Space is T3|Metric Space is $T_3$]].
{{qed}}
[[Category:Real Number Line satisfies all Separation Axioms]]
[[Category:Real Number Line with Euclid... | Real Number Line with Euclidean Topology is T3 | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_T3 | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_T3 | [
"Real Number Line satisfies all Separation Axioms",
"Real Number Line with Euclidean Topology",
"Examples of T3 Spaces"
] | [
"Definition:T3 Space"
] | [
"Real Number Line is Metric Space",
"Definition:Metric Space",
"Metric Space is T3",
"Category:Real Number Line satisfies all Separation Axioms",
"Category:Real Number Line with Euclidean Topology",
"Category:Examples of T3 Spaces"
] |
proofwiki-23292 | Real Number Line with Euclidean Topology is T4 | :$\struct {\R, \tau_d}$ is a $T_4$ space. | From Real Number Line is Metric Space we have that $\struct {\R, \tau_d}$ is an example of a metric space.
The result follows from Metric Space is $T_4$.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of T4 Spaces
lm69reqvk0umit8cyop... | :$\struct {\R, \tau_d}$ is a [[Definition:T4 Space|$T_4$ space]]. | From [[Real Number Line is Metric Space]] we have that $\struct {\R, \tau_d}$ is an example of a [[Definition:Metric Space|metric space]].
The result follows from [[Metric Space is T4|Metric Space is $T_4$]].
{{qed}}
[[Category:Real Number Line satisfies all Separation Axioms]]
[[Category:Real Number Line with Euclid... | Real Number Line with Euclidean Topology is T4 | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_T4 | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_T4 | [
"Real Number Line satisfies all Separation Axioms",
"Real Number Line with Euclidean Topology",
"Examples of T4 Spaces"
] | [
"Definition:T4 Space"
] | [
"Real Number Line is Metric Space",
"Definition:Metric Space",
"Metric Space is T4",
"Category:Real Number Line satisfies all Separation Axioms",
"Category:Real Number Line with Euclidean Topology",
"Category:Examples of T4 Spaces"
] |
proofwiki-23293 | Real Number Line with Euclidean Topology is T5 | :$\struct {\R, \tau_d}$ is a $T_5$ space | From Real Number Line is Metric Space we have that $\struct {\R, \tau_d}$ is an example of a metric space.
The result follows from Metric Space is $T_5$.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of T5 Spaces
k7vm1tcvziwgzy6jxn9... | :$\struct {\R, \tau_d}$ is a [[Definition:T5 Space|$T_5$ space]] | From [[Real Number Line is Metric Space]] we have that $\struct {\R, \tau_d}$ is an example of a [[Definition:Metric Space|metric space]].
The result follows from [[Metric Space is T5|Metric Space is $T_5$]].
{{qed}}
[[Category:Real Number Line satisfies all Separation Axioms]]
[[Category:Real Number Line with Euclid... | Real Number Line with Euclidean Topology is T5 | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_T5 | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_T5 | [
"Real Number Line satisfies all Separation Axioms",
"Real Number Line with Euclidean Topology",
"Examples of T5 Spaces"
] | [
"Definition:T5 Space"
] | [
"Real Number Line is Metric Space",
"Definition:Metric Space",
"Metric Space is T5",
"Category:Real Number Line satisfies all Separation Axioms",
"Category:Real Number Line with Euclidean Topology",
"Category:Examples of T5 Spaces"
] |
proofwiki-23294 | Real Number Line with Euclidean Topology is Perfectly T4 | :$\struct {\R, \tau_d}$ is a perfectly $T_4$ space | From Real Number Line is Metric Space we have that $\struct {\R, \tau_d}$ is an example of a metric space.
The result follows from Metric Space is Perfectly $T_4$.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of Perfectly T4 Spaces... | :$\struct {\R, \tau_d}$ is a [[Definition:Perfectly T4 Space|perfectly $T_4$ space]] | From [[Real Number Line is Metric Space]] we have that $\struct {\R, \tau_d}$ is an example of a [[Definition:Metric Space|metric space]].
The result follows from [[Metric Space is Perfectly T4|Metric Space is Perfectly $T_4$]].
{{qed}}
[[Category:Real Number Line satisfies all Separation Axioms]]
[[Category:Real Num... | Real Number Line with Euclidean Topology is Perfectly T4 | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_Perfectly_T4 | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_Perfectly_T4 | [
"Real Number Line satisfies all Separation Axioms",
"Real Number Line with Euclidean Topology",
"Examples of Perfectly T4 Spaces"
] | [
"Definition:Perfectly T4 Space"
] | [
"Real Number Line is Metric Space",
"Definition:Metric Space",
"Metric Space is Perfectly T4",
"Category:Real Number Line satisfies all Separation Axioms",
"Category:Real Number Line with Euclidean Topology",
"Category:Examples of Perfectly T4 Spaces"
] |
proofwiki-23295 | Real Number Line with Euclidean Topology is Fully T4 | :$\struct {\R, \tau_d}$ is a fully $T_4$ space. | From Real Number Line is Metric Space we have that $\struct {\R, \tau_d}$ is an example of a metric space.
The result follows from Metric Space is Fully $T_4$.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of Fully T4 Spaces
qsuy1ss... | :$\struct {\R, \tau_d}$ is a [[Definition:Fully T4 Space|fully $T_4$ space]]. | From [[Real Number Line is Metric Space]] we have that $\struct {\R, \tau_d}$ is an example of a [[Definition:Metric Space|metric space]].
The result follows from [[Metric Space is Fully T4|Metric Space is Fully $T_4$]].
{{qed}}
[[Category:Real Number Line satisfies all Separation Axioms]]
[[Category:Real Number Line... | Real Number Line with Euclidean Topology is Fully T4 | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_Fully_T4 | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_Fully_T4 | [
"Real Number Line satisfies all Separation Axioms",
"Real Number Line with Euclidean Topology",
"Examples of Fully T4 Spaces"
] | [
"Definition:Fully T4 Space"
] | [
"Real Number Line is Metric Space",
"Definition:Metric Space",
"Metric Space is Fully T4",
"Category:Real Number Line satisfies all Separation Axioms",
"Category:Real Number Line with Euclidean Topology",
"Category:Examples of Fully T4 Spaces"
] |
proofwiki-23296 | Real Number Line with Euclidean Topology is Regular | :$\struct {\R, \tau_d}$ is a regular space | From Real Number Line is Metric Space we have that $\struct {\R, \tau_d}$ is an example of a metric space.
The result follows from Metric Space is Regular.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of Regular Spaces
kdkvnn7hqh3w... | :$\struct {\R, \tau_d}$ is a [[Definition:Regular Space|regular space]] | From [[Real Number Line is Metric Space]] we have that $\struct {\R, \tau_d}$ is an example of a [[Definition:Metric Space|metric space]].
The result follows from [[Metric Space is Regular]].
{{qed}}
[[Category:Real Number Line satisfies all Separation Axioms]]
[[Category:Real Number Line with Euclidean Topology]]
[[... | Real Number Line with Euclidean Topology is Regular | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_Regular | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_Regular | [
"Real Number Line satisfies all Separation Axioms",
"Real Number Line with Euclidean Topology",
"Examples of Regular Spaces"
] | [
"Definition:Regular Space"
] | [
"Real Number Line is Metric Space",
"Definition:Metric Space",
"Metric Space is Regular",
"Category:Real Number Line satisfies all Separation Axioms",
"Category:Real Number Line with Euclidean Topology",
"Category:Examples of Regular Spaces"
] |
proofwiki-23297 | Real Number Line with Euclidean Topology is Urysohn | :$\struct {\R, \tau_d}$ is an Urysohn space | From Real Number Line is Metric Space we have that $\struct {\R, \tau_d}$ is an example of a metric space.
The result follows from Metric Space is Urysohn.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of Urysohn Spaces
pgug638hzoo0... | :$\struct {\R, \tau_d}$ is an [[Definition:Urysohn Space|Urysohn space]] | From [[Real Number Line is Metric Space]] we have that $\struct {\R, \tau_d}$ is an example of a [[Definition:Metric Space|metric space]].
The result follows from [[Metric Space is Urysohn]].
{{qed}}
[[Category:Real Number Line satisfies all Separation Axioms]]
[[Category:Real Number Line with Euclidean Topology]]
[[... | Real Number Line with Euclidean Topology is Urysohn | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_Urysohn | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_Urysohn | [
"Real Number Line satisfies all Separation Axioms",
"Real Number Line with Euclidean Topology",
"Examples of Urysohn Spaces"
] | [
"Definition:Urysohn Space"
] | [
"Real Number Line is Metric Space",
"Definition:Metric Space",
"Metric Space is Urysohn",
"Category:Real Number Line satisfies all Separation Axioms",
"Category:Real Number Line with Euclidean Topology",
"Category:Examples of Urysohn Spaces"
] |
proofwiki-23298 | Real Number Line with Euclidean Topology is Completely Regular | :$\struct {\R, \tau_d}$ is a completely regular space | From Real Number Line is Metric Space we have that $\struct {\R, \tau_d}$ is an example of a metric space.
The result follows from Metric Space is Completely Regular.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of Completely Regul... | :$\struct {\R, \tau_d}$ is a [[Definition:Completely Regular Space|completely regular space]] | From [[Real Number Line is Metric Space]] we have that $\struct {\R, \tau_d}$ is an example of a [[Definition:Metric Space|metric space]].
The result follows from [[Metric Space is Completely Regular]].
{{qed}}
[[Category:Real Number Line satisfies all Separation Axioms]]
[[Category:Real Number Line with Euclidean To... | Real Number Line with Euclidean Topology is Completely Regular | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_Completely_Regular | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_Completely_Regular | [
"Real Number Line satisfies all Separation Axioms",
"Real Number Line with Euclidean Topology",
"Examples of Completely Regular Spaces"
] | [
"Definition:Completely Regular Space"
] | [
"Real Number Line is Metric Space",
"Definition:Metric Space",
"Metric Space is Completely Regular",
"Category:Real Number Line satisfies all Separation Axioms",
"Category:Real Number Line with Euclidean Topology",
"Category:Examples of Completely Regular Spaces"
] |
proofwiki-23299 | Real Number Line with Euclidean Topology is Completely Normal | :$\struct {\R, \tau_d}$ is a completely normal space | From Real Number Line is Metric Space we have that $\struct {\R, \tau_d}$ is an example of a metric space.
The result follows from Metric Space is Completely Normal.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of Completely Normal... | :$\struct {\R, \tau_d}$ is a [[Definition:Completely Normal Space|completely normal space]] | From [[Real Number Line is Metric Space]] we have that $\struct {\R, \tau_d}$ is an example of a [[Definition:Metric Space|metric space]].
The result follows from [[Metric Space is Completely Normal]].
{{qed}}
[[Category:Real Number Line satisfies all Separation Axioms]]
[[Category:Real Number Line with Euclidean Top... | Real Number Line with Euclidean Topology is Completely Normal | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_Completely_Normal | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_Completely_Normal | [
"Real Number Line satisfies all Separation Axioms",
"Real Number Line with Euclidean Topology",
"Examples of Completely Normal Spaces"
] | [
"Definition:Completely Normal Space"
] | [
"Real Number Line is Metric Space",
"Definition:Metric Space",
"Metric Space is Completely Normal",
"Category:Real Number Line satisfies all Separation Axioms",
"Category:Real Number Line with Euclidean Topology",
"Category:Examples of Completely Normal Spaces"
] |
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