Question

Suppose that $X$ is homotopy-equivalent to a one-point space, then each point of $X$ is a deformation retract of $X$.

score 622 · Answer 1 · 2026-04-03 04:46:02

622

Views

Suppose that $X$ is homotopy-equivalent to a one-point space, then each point of $X$ is a deformation retract of $X$.

Published on 03 Apr 2026 - 4:46

score 166 · Answer 2 · 2026-04-11 20:10:50

166

Views

How to prove a region remains connected if one point is excluded?

Published on 11 Apr 2026 - 20:10

score 649 · Answer 3 · 2026-03-25 11:16:34

649

Views

Prob. 1, Sec. 25, in Munkres' TOPOLOGY, 2nd ed: The components and the path components of $\mathbb{R}$ with lower limit topology

Published on 25 Mar 2026 - 11:16

score 66 · Answer 4 · 2026-03-29 20:11:33

66

Views

If $C$ is a closed subset of a metric space, then $\bar{B}(C, r)$ is a closed subset also?

Published on 29 Mar 2026 - 20:11

score 863 · Answer 5 · 2026-03-25 18:51:15

863

Views

Countable Dense Set not a $G_{\delta}$ set? Where's the contradiction?

Published on 25 Mar 2026 - 18:51

score 426 · Answer 6 · 2026-04-11 20:02:16

426

Views

Prob. 2(b), Sec. 25, in Munkres' TOPOLOGY, 2nd ed: The iff-condition for two points to be in the same component of $\mathbb{R}^\omega$

Published on 11 Apr 2026 - 20:02

score 101 · Answer 7 · 2026-03-25 03:20:40

101

Views

Number of homotopically inequivalent loops on a surface

Published on 25 Mar 2026 - 3:20

score 52 · Answer 8 · 2018-07-28 01:54:15

52

Views

$(-\sqrt 2, \sqrt 2) \cap \mathbb Q$ in order topology. What will be it's bases?

Published on 28 Jul 2018 - 1:54

score 2.5k · Answer 9 · 2026-03-25 04:43:49

2.5k

Views

Suppose $f$ holomorphic and its image is a subset of the unit circle. Then show f is constant.

Published on 25 Mar 2026 - 4:43

score 2.9k · Answer 10 · 2018-07-28 11:44:26

2.9k

Views

Is $\operatorname{int}(A\cup B)=\operatorname{int}(A)\cup\operatorname{int}(B)$?

Published on 28 Jul 2018 - 11:44

score 702 · Answer 11 · 2026-03-28 13:21:20

702

Views

Show that the disjoint union is the coproduct in the category of topological spaces

Published on 28 Mar 2026 - 13:21

score 270 · Answer 12 · 2026-04-02 01:54:49

270

Views

What do you call this property involving a function between two complete metric spaces?

Published on 02 Apr 2026 - 1:54

score 118 · Answer 13 · 2026-04-11 10:52:26

118

Views

Are the path components Borel sets?

Published on 11 Apr 2026 - 10:52

score 30 · Answer 14 · 2026-03-25 03:24:05

30

Views

Number of homotopically inequivalent loops that pairwise intersect at most once

Published on 25 Mar 2026 - 3:24

score 62 · Answer 15 · 2026-04-03 12:38:10

62

Views

On a certain topology on the set of continuous functions between topological spaces

Published on 03 Apr 2026 - 12:38

List Question

Suppose that $X$ is homotopy-equivalent to a one-point space, then each point of $X$ is a deformation retract of $X$.

How to prove a region remains connected if one point is excluded?

Prob. 1, Sec. 25, in Munkres' TOPOLOGY, 2nd ed: The components and the path components of $\mathbb{R}$ with lower limit topology

If $C$ is a closed subset of a metric space, then $\bar{B}(C, r)$ is a closed subset also?

Countable Dense Set not a $G_{\delta}$ set? Where's the contradiction?

Prob. 2(b), Sec. 25, in Munkres' TOPOLOGY, 2nd ed: The iff-condition for two points to be in the same component of $\mathbb{R}^\omega$

Number of homotopically inequivalent loops on a surface

$(-\sqrt 2, \sqrt 2) \cap \mathbb Q$ in order topology. What will be it's bases?

Suppose $f$ holomorphic and its image is a subset of the unit circle. Then show f is constant.

Is $\operatorname{int}(A\cup B)=\operatorname{int}(A)\cup\operatorname{int}(B)$?

Show that the disjoint union is the coproduct in the category of topological spaces

What do you call this property involving a function between two complete metric spaces?

Are the path components Borel sets?

Number of homotopically inequivalent loops that pairwise intersect at most once

On a certain topology on the set of continuous functions between topological spaces

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