Question

Does any isomorphism between $\pi_1(X,x_0)$ and $\pi_1(Y,y_0)$ always induce a homeomorphism between $(X,x_0)$ and $(Y,y_0)$?

score 368 · Answer 1 · 2026-03-26 01:23:33

368

Views

Does any isomorphism between $\pi_1(X,x_0)$ and $\pi_1(Y,y_0)$ always induce a homeomorphism between $(X,x_0)$ and $(Y,y_0)$?

Published on 26 Mar 2026 - 1:23

score 96 · Answer 2 · 2026-04-09 03:36:38

96

Views

$\mathbb C^{\infty}\setminus\{0\}$ retracts on $S^{\infty}$

Published on 09 Apr 2026 - 3:36

score 666 · Answer 3 · 2026-04-12 23:36:14

666

Views

Topology definition: finite intersections vs. infinite unions

Published on 12 Apr 2026 - 23:36

score 145 · Answer 4 · 2026-03-25 09:48:46

145

Views

Given $k$ points in $n$-dimensional space, is there always a continuous $n-1$ surface that can divide the points into two arbitrary groups?

Published on 25 Mar 2026 - 9:48

score 1k · Answer 5 · 2026-04-11 22:27:04

1k

Views

Example of a CW complex that is not a $\Delta$-complex?

Published on 11 Apr 2026 - 22:27

score 299 · Answer 6 · 2018-08-12 01:36:38

299

Views

Show that a proper locally invertible map is surjective

Published on 12 Aug 2018 - 1:36

score 99 · Answer 7 · 2026-03-25 04:59:50

99

Views

Prove path independence of $\int_{\gamma} f$ with weaker conditions on $f$

Published on 25 Mar 2026 - 4:59

score 3.7k · Answer 8 · 2026-04-11 08:07:02

3.7k

Views

Continuous bijection is homeomorphism

Published on 11 Apr 2026 - 8:07

score 850 · Answer 9 · 2026-04-12 21:13:35

850

Views

every open set in R is the union of an at most countable collection of disjoint segments

Published on 12 Apr 2026 - 21:13

score 1.3k · Answer 10 · 2026-04-11 14:26:17

1.3k

Views

Difference between closed under countable union and arbitrary union

Published on 11 Apr 2026 - 14:26

score 463 · Answer 11 · 2026-03-29 03:53:27

463

Views

Why $\mathbb R$ with the metric $d(x,y)=\min\{|x-y|,1\}$ is not compact?

Published on 29 Mar 2026 - 3:53

score 757 · Answer 12 · 2018-08-12 17:41:43

757

Views

Every point of the unit circle is a limit point of $x_k = \left(1+\frac{1}{2^k}\right)\begin{bmatrix} \cos k\\\sin k\\ \end{bmatrix}$

Published on 12 Aug 2018 - 17:41

score 97 · Answer 13 · 2026-03-26 02:56:02

97

Views

Prove that $\{x\in \Bbb{R^n} :\;f(x)\leq k\}$ is compact

Published on 26 Mar 2026 - 2:56

score 49 · Answer 14 · 2018-08-12 19:48:21

49

Views

What are these Moebius-esque strips called?

Published on 12 Aug 2018 - 19:48

score 513 · Answer 15 · 2026-04-09 01:59:27

513

Views

How to show that $C^\infty_0$ is not dense in $L^p_{weak} (\mathbb{R}^n)$?

Published on 09 Apr 2026 - 1:59

List Question

Does any isomorphism between $\pi_1(X,x_0)$ and $\pi_1(Y,y_0)$ always induce a homeomorphism between $(X,x_0)$ and $(Y,y_0)$?

$\mathbb C^{\infty}\setminus\{0\}$ retracts on $S^{\infty}$

Topology definition: finite intersections vs. infinite unions

Given $k$ points in $n$-dimensional space, is there always a continuous $n-1$ surface that can divide the points into two arbitrary groups?

Example of a CW complex that is not a $\Delta$-complex?

Show that a proper locally invertible map is surjective

Prove path independence of $\int_{\gamma} f$ with weaker conditions on $f$

Continuous bijection is homeomorphism

every open set in R is the union of an at most countable collection of disjoint segments

Difference between closed under countable union and arbitrary union

Why $\mathbb R$ with the metric $d(x,y)=\min\{|x-y|,1\}$ is not compact?

Every point of the unit circle is a limit point of $x_k = \left(1+\frac{1}{2^k}\right)\begin{bmatrix} \cos k\\\sin k\\ \end{bmatrix}$

Prove that $\{x\in \Bbb{R^n} :\;f(x)\leq k\}$ is compact

What are these Moebius-esque strips called?

How to show that $C^\infty_0$ is not dense in $L^p_{weak} (\mathbb{R}^n)$?

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