Question

Let $X \subset \mathbb{R}$ be $G_{\delta}$ is homeomorphic to $\mathcal{N}$.

score 162 · Answer 1 · 2026-02-23 02:56:51

162

Views

Let $X \subset \mathbb{R}$ be $G_{\delta}$ is homeomorphic to $\mathcal{N}$.

Published on 23 Feb 2026 - 2:56

score 545 · Answer 2 · 2019-09-22 23:38:38

545

Views

Example where $A$ and $B$ closed, $A+B$ not an $F_{\sigma}$

Published on 22 Sep 2019 - 23:38

score 38 · Answer 3 · 2019-09-23 03:27:32

38

Views

Maximum Value Theorem proof in a eucledian sapce

Published on 23 Sep 2019 - 3:27

score 75 · Answer 4 · 2019-09-23 05:03:16

75

Views

A question on the definition of a compact set

Published on 23 Sep 2019 - 5:03

score 125 · Answer 5 · 2019-09-24 03:20:31

125

Views

Assume that $f : A \to A$ is a function such that $\forall x, y \in A,\|f(y)-f(x)\|=\|y-x\|$. Prove that $f(A)= A$

Published on 24 Sep 2019 - 3:20

score 80 · Answer 6 · 2026-03-26 06:29:03

80

Views

Show that $\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$ is complete and compact with respect to given metric.

Published on 26 Mar 2026 - 6:29

score 26 · Answer 7 · 2019-09-25 07:46:54

26

Views

Show that the function $s$ is continuous

Published on 25 Sep 2019 - 7:46

score 46 · Answer 8 · 2019-09-25 14:02:49

46

Views

Compactness of the set.

Published on 25 Sep 2019 - 14:02

score 1.7k · Answer 9 · 2019-09-25 18:16:38

1.7k

Views

Let $X$ and $Y$ be metric spaces, $X$ compact, and $T: X \to Y$ bijective and continuous. Show that $T$ is a homeomorphism.

Published on 25 Sep 2019 - 18:16

score 178 · Answer 10 · 2019-09-26 14:03:28

178

Views

Is the set of matrix with bounded entries are compact set?

Published on 26 Sep 2019 - 14:03

score 35 · Answer 11 · 2019-09-27 13:07:57

35

Views

Prove $E_r = \{z \in \mathbb{C} \text{ }|\text{ } |z - w| \leq r \text{ for some } w \in E \}$ is compact, where $E$ is compact.

Published on 27 Sep 2019 - 13:07

score 336 · Answer 12 · 2019-09-27 15:09:40

336

Views

Heine borel theorem doesn't hold

Published on 27 Sep 2019 - 15:09

score 204 · Answer 13 · 2026-03-25 23:39:06

204

Views

Lebesgue measure of the projections of a compact set

Published on 25 Mar 2026 - 23:39

score 715 · Answer 14 · 2026-03-26 14:19:22

715

Views

Does a monotonically increasing function attain a maximum on a compact subset of $\mathbb{R}$? Prove that it does, or give an example against.

Published on 26 Mar 2026 - 14:19

score 179 · Answer 15 · 2019-09-30 16:51:23

179

Views

Intuition for why $(a, b)$ is not compact, but $[a, b]$ is compact, where $[a, b]$ is contained by some metric space $(X, d)$.

Published on 30 Sep 2019 - 16:51

List Question

Let $X \subset \mathbb{R}$ be $G_{\delta}$ is homeomorphic to $\mathcal{N}$.

Example where $A$ and $B$ closed, $A+B$ not an $F_{\sigma}$

Maximum Value Theorem proof in a eucledian sapce

A question on the definition of a compact set

Assume that $f : A \to A$ is a function such that $\forall x, y \in A,\|f(y)-f(x)\|=\|y-x\|$. Prove that $f(A)= A$

Show that $\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$ is complete and compact with respect to given metric.

Show that the function $s$ is continuous

Compactness of the set.

Let $X$ and $Y$ be metric spaces, $X$ compact, and $T: X \to Y$ bijective and continuous. Show that $T$ is a homeomorphism.

Is the set of matrix with bounded entries are compact set?

Prove $E_r = \{z \in \mathbb{C} \text{ }|\text{ } |z - w| \leq r \text{ for some } w \in E \}$ is compact, where $E$ is compact.

Heine borel theorem doesn't hold

Lebesgue measure of the projections of a compact set

Does a monotonically increasing function attain a maximum on a compact subset of $\mathbb{R}$? Prove that it does, or give an example against.

Intuition for why $(a, b)$ is not compact, but $[a, b]$ is compact, where $[a, b]$ is contained by some metric space $(X, d)$.

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