Question

Show that if $X$ has the discrete topology, then $X$ is totally disconnected. Does the converse hold?

score 3.6k · Answer 1 · 2018-04-06 12:20:06

3.6k

Views

Show that if $X$ has the discrete topology, then $X$ is totally disconnected. Does the converse hold?

Published on 06 Apr 2018 - 12:20

score 97 · Answer 2 · 2026-03-27 20:12:49

97

Views

Compactness of spaces

Published on 27 Mar 2026 - 20:12

score 46 · Answer 3 · 2026-03-25 18:07:51

46

Views

Stability of "separating set" under continuous maps that preserve boundaries

Published on 25 Mar 2026 - 18:07

score 44 · Answer 4 · 2018-04-06 19:00:48

44

Views

Inclusions of closed sets

Published on 06 Apr 2018 - 19:00

score 112 · Answer 5 · 2026-03-25 19:43:01

112

Views

Let $P = \{(x,y) \in \mathbb R^2 : x^2+y^2<1 \}$ and $Q = \{(1,x) : x \in \mathbb R\}$. Is $X=P \cup Q$ connected?

Published on 25 Mar 2026 - 19:43

score 581 · Answer 6 · 2026-03-25 16:01:06

581

Views

For any graph, is there a topological space where it can be drawn/embedded?

Published on 25 Mar 2026 - 16:01

score 132 · Answer 7 · 2026-03-25 19:25:17

132

Views

If $x \in \overline{A}$ implies $x$ is the limit of a sequence in $A$ for all $A \subset X$, is $X$ first-countable?

Published on 25 Mar 2026 - 19:25

score 52 · Answer 8 · 2026-03-25 06:04:56

52

Views

$b)$ what happen if $X$ equals $[0,1) $or $(0,1)$?

Published on 25 Mar 2026 - 6:04

score 951 · Answer 9 · 2026-03-27 14:39:34

951

Views

Canonical projection is open - Projective Space

Published on 27 Mar 2026 - 14:39

score 219 · Answer 10 · 2018-04-07 17:49:23

219

Views

Consider the following set in the dictionary order.?

Published on 07 Apr 2018 - 17:49

score 112 · Answer 11 · 2026-03-25 19:43:01

112

Views

Why is $\operatorname{Bd}(A) = \{0, 1\}$ not connected and $\operatorname{int}(\Bbb Q) = \varnothing$ connected?

Published on 25 Mar 2026 - 19:43

score 974 · Answer 12 · 2026-03-25 04:44:17

974

Views

Show that the product topology on $X\times Y$ is the same as the disjoint union topology on $\bigsqcup_{y\in Y}X$.

Published on 25 Mar 2026 - 4:44

score 182 · Answer 13 · 2018-04-07 20:09:40

182

Views

Show that if $U$ is an open connected subspace of $\mathbb{R}^2$, then $U$ is path connected.

Published on 07 Apr 2018 - 20:09

score 149 · Answer 14 · 2018-04-08 04:25:09

149

Views

What topological properties need a space have to be topologically euclidean?

Published on 08 Apr 2018 - 4:25

score 58 · Answer 15 · 2026-03-25 12:34:35

58

Views

Compact subset of cantor set meeting all but one dense subset

Published on 25 Mar 2026 - 12:34

List Question

Show that if $X$ has the discrete topology, then $X$ is totally disconnected. Does the converse hold?

Compactness of spaces

Stability of "separating set" under continuous maps that preserve boundaries

Inclusions of closed sets

Let $P = \{(x,y) \in \mathbb R^2 : x^2+y^2<1 \}$ and $Q = \{(1,x) : x \in \mathbb R\}$. Is $X=P \cup Q$ connected?

For any graph, is there a topological space where it can be drawn/embedded?

If $x \in \overline{A}$ implies $x$ is the limit of a sequence in $A$ for all $A \subset X$, is $X$ first-countable?

$b)$ what happen if $X$ equals $[0,1) $or $(0,1)$?

Canonical projection is open - Projective Space

Consider the following set in the dictionary order.?

Why is $\operatorname{Bd}(A) = \{0, 1\}$ not connected and $\operatorname{int}(\Bbb Q) = \varnothing$ connected?

Show that the product topology on $X\times Y$ is the same as the disjoint union topology on $\bigsqcup_{y\in Y}X$.

Show that if $U$ is an open connected subspace of $\mathbb{R}^2$, then $U$ is path connected.

What topological properties need a space have to be topologically euclidean?

Compact subset of cantor set meeting all but one dense subset

Trending Questions

Popular # Hahtags

Popular Questions