Question

If $f$ is continuous $\displaystyle\lim_{x \to \infty}f(x)=\infty$ iff $f^{-1}(K)$ is compact for every compact $K \subset \mathbb{R}^{n}$.

score 193 · Answer 1 · 2026-04-14 04:25:58

193

Views

If $f$ is continuous $\displaystyle\lim_{x \to \infty}f(x)=\infty$ iff $f^{-1}(K)$ is compact for every compact $K \subset \mathbb{R}^{n}$.

Published on 14 Apr 2026 - 4:25

score 553 · Answer 2 · 2026-04-12 14:32:31

553

Views

Why set of integer under indiscrete topology is compact?

Published on 12 Apr 2026 - 14:32

score 146 · Answer 3 · 2026-05-11 02:32:02

146

Views

Limit point compact uniform space

Published on 11 May 2026 - 2:32

score 422 · Answer 4 · 2026-04-15 18:50:00

422

Views

If $f:X\rightarrow X$ is continuous and $(X,d)$ is compact, show there is $\epsilon>0$ such that $d(x,f(x))>\epsilon$ for all $x\in X$.

Published on 15 Apr 2026 - 18:50

score 144 · Answer 5 · 2026-04-11 15:24:09

144

Views

Infinite dimensional Banach spaces and sequences

Published on 11 Apr 2026 - 15:24

score 127 · Answer 6 · 2026-05-10 16:53:20

127

Views

What is the compactness uniqueness argument

Published on 10 May 2026 - 16:53

score 32 · Answer 7 · 2026-05-10 20:08:26

32

Views

Why can('t) I map this closed subset of a product of compact sets to a non-compact set?

Published on 10 May 2026 - 20:08

score 979 · Answer 8 · 2026-05-14 13:27:01

979

Views

If $X$ is compact and Hausdorff, and $f:X\rightarrow Y$ is continuous, closed, and surjective, prove $Y$ is Hausdorff.

Published on 14 May 2026 - 13:27

score 601 · Answer 9 · 2026-05-14 13:21:03

601

Views

Difference between sequentially compact spaces and closed sets in $\mathbb{C}$

Published on 14 May 2026 - 13:21

score 83 · Answer 10 · 2026-05-14 14:23:30

83

Views

countably compact and compact

Published on 14 May 2026 - 14:23

score 444 · Answer 11 · 2026-05-11 09:58:57

444

Views

Let $q\colon E\to X$ be a covering map. If $E$ is compact then $X$ is compact and $q$ is a finite-sheeted covering.

Published on 11 May 2026 - 9:58

score 278 · Answer 12 · 2026-04-09 10:01:33

278

Views

Equicontinuous homeomorphism and compact metric

Published on 09 Apr 2026 - 10:01

score 722 · Answer 13 · 2026-05-16 01:40:21

722

Views

Compact set always contains its sup and inf

Published on 16 May 2026 - 1:40

score 1.1k · Answer 14 · 2026-05-14 16:21:10

1.1k

Views

Is $ C ^ 1 [ 0 , 1 ] $ compact? Is the closure of the unit ball of $ C ^ 1 [ 0 , 1 ] $ in $ C [ 0 , 1 ] $ compact?

Published on 14 May 2026 - 16:21

score 295 · Answer 15 · 2026-05-14 17:12:36

295

Views

Compact spaces in which any closed set can be partitioned into finitely many closed sets whose clopen subsets extend to the whole space

Published on 14 May 2026 - 17:12

List Question

If $f$ is continuous $\displaystyle\lim_{x \to \infty}f(x)=\infty$ iff $f^{-1}(K)$ is compact for every compact $K \subset \mathbb{R}^{n}$.

Why set of integer under indiscrete topology is compact?

Limit point compact uniform space

If $f:X\rightarrow X$ is continuous and $(X,d)$ is compact, show there is $\epsilon>0$ such that $d(x,f(x))>\epsilon$ for all $x\in X$.

Infinite dimensional Banach spaces and sequences

What is the compactness uniqueness argument

Why can('t) I map this closed subset of a product of compact sets to a non-compact set?

If $X$ is compact and Hausdorff, and $f:X\rightarrow Y$ is continuous, closed, and surjective, prove $Y$ is Hausdorff.

Difference between sequentially compact spaces and closed sets in $\mathbb{C}$

countably compact and compact

Let $q\colon E\to X$ be a covering map. If $E$ is compact then $X$ is compact and $q$ is a finite-sheeted covering.

Equicontinuous homeomorphism and compact metric

Compact set always contains its sup and inf

Is $ C ^ 1 [ 0 , 1 ] $ compact? Is the closure of the unit ball of $ C ^ 1 [ 0 , 1 ] $ in $ C [ 0 , 1 ] $ compact?

Compact spaces in which any closed set can be partitioned into finitely many closed sets whose clopen subsets extend to the whole space

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