Question

Topology, Hausdorff and Fréchet space

score 120 · Answer 1 · 2022-01-04 18:46:03

120

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Topology, Hausdorff and Fréchet space

Published on 04 Jan 2022 - 18:46

score 74 · Answer 2 · 2022-01-07 13:06:36

74

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Does $T_3$ imply that the topological space is zero-dimensional?

Published on 07 Jan 2022 - 13:06

score 228 · Answer 3 · 2022-01-08 11:40:08

228

Views

Let $X,Y$ be Hausdorff and $f,g : X \to Y$ continuous. Let $A \subset X$ be dense and suppose that $f\mid_A = g\mid_A$. Show that $f=g$.

Published on 08 Jan 2022 - 11:40

score 132 · Answer 4 · 2022-01-13 16:52:02

132

Views

Let $X$ be a Hausdorff space and $A \subset X$ compact. Show that $A$ is closed.

Published on 13 Jan 2022 - 16:52

score 514 · Answer 5 · 2022-01-15 08:09:26

514

Views

Prove or disprove : In a topological space $(X,\tau)$ if every compact subsets $K\subset X$ are closed then $(X, \tau) $ is hausdorff.

Published on 15 Jan 2022 - 8:09

score 195 · Answer 6 · 2022-01-16 20:39:14

195

Views

Is the characterization of Hausdorff spaces in terms of ultrafilter convergence equivalent to the ultrafilter lemma?

Published on 16 Jan 2022 - 20:39

score 42 · Answer 7 · 2022-01-19 15:33:20

42

Views

$X$ is $T_1$ implies $f(X)$ is $T_1$

Published on 19 Jan 2022 - 15:33

score 58 · Answer 8 · 2022-01-22 16:26:27

58

Views

Any neighbourhood of a closed decreasing sequence in a compact space must contain one of the sets: why is the space required also to be Hausdorff?

Published on 22 Jan 2022 - 16:26

score 71 · Answer 9 · 2022-01-27 15:21:17

Let $D$ be a partition of a space $X$ with the quotient topology. Show that $D$ is $T_1$ if and only if the members of $D$ are closed.

score 147 · Answer 10 · 2022-01-27 18:10:58

147

Views

Show that a Hausdorff space $X$ is regular if and only if $X/A$ is Hausdorff for all $A$ closed in $X$.

Published on 27 Jan 2022 - 18:10

score 81 · Answer 11 · 2022-01-28 14:34:19

81

Views

Proof verification on Munkres Chapter 4, Section 31, Problem 7

Published on 28 Jan 2022 - 14:34

score 49 · Answer 12 · 2022-01-30 16:17:33

49

Views

Separating points of closed countable discrete subspace of a $T_4$ space with pairwise disjoint open sets

Published on 30 Jan 2022 - 16:17

score 81 · Answer 13 · 2022-01-30 19:22:15

81

Views

Show that a finite $T_0$-space has at least one point which is a closed subset

Published on 30 Jan 2022 - 19:22

score 194 · Answer 14 · 2022-02-02 00:49:39

194

Views

Completely Hausdorff space is Urysohn

Published on 02 Feb 2022 - 0:49

score 161 · Answer 15 · 2022-02-02 12:35:50

161

Views

Can infinite $T_0$ space have less open sets than points?

Published on 02 Feb 2022 - 12:35

List Question

Topology, Hausdorff and Fréchet space

Does $T_3$ imply that the topological space is zero-dimensional?

Let $X,Y$ be Hausdorff and $f,g : X \to Y$ continuous. Let $A \subset X$ be dense and suppose that $f\mid_A = g\mid_A$. Show that $f=g$.

Let $X$ be a Hausdorff space and $A \subset X$ compact. Show that $A$ is closed.

Prove or disprove : In a topological space $(X,\tau)$ if every compact subsets $K\subset X$ are closed then $(X, \tau) $ is hausdorff.

Is the characterization of Hausdorff spaces in terms of ultrafilter convergence equivalent to the ultrafilter lemma?

$X$ is $T_1$ implies $f(X)$ is $T_1$

Any neighbourhood of a closed decreasing sequence in a compact space must contain one of the sets: why is the space required also to be Hausdorff?

Let $D$ be a partition of a space $X$ with the quotient topology. Show that $D$ is $T_1$ if and only if the members of $D$ are closed.

Show that a Hausdorff space $X$ is regular if and only if $X/A$ is Hausdorff for all $A$ closed in $X$.

Proof verification on Munkres Chapter 4, Section 31, Problem 7

Separating points of closed countable discrete subspace of a $T_4$ space with pairwise disjoint open sets

Show that a finite $T_0$-space has at least one point which is a closed subset

Completely Hausdorff space is Urysohn

Can infinite $T_0$ space have less open sets than points?

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