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 191 · Answer 1 · 2019-01-06 03:49:35

191

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 06 Jan 2019 - 3:49

score 550 · Answer 2 · 2019-01-06 10:45:43

550

Views

Why set of integer under indiscrete topology is compact?

Published on 06 Jan 2019 - 10:45

score 143 · Answer 3 · 2026-03-26 23:47:19

143

Views

Limit point compact uniform space

Published on 26 Mar 2026 - 23:47

score 419 · Answer 4 · 2019-01-07 14:26:26

419

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 07 Jan 2019 - 14:26

score 142 · Answer 5 · 2026-03-28 04:33:19

142

Views

Infinite dimensional Banach spaces and sequences

Published on 28 Mar 2026 - 4:33

score 124 · Answer 6 · 2026-03-27 09:15:38

124

Views

What is the compactness uniqueness argument

Published on 27 Mar 2026 - 9:15

score 29 · Answer 7 · 2026-03-25 04:43:05

29

Views

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

Published on 25 Mar 2026 - 4:43

score 975 · Answer 8 · 2019-01-13 00:43:50

975

Views

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

Published on 13 Jan 2019 - 0:43

score 597 · Answer 9 · 2019-01-13 12:50:04

597

Views

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

Published on 13 Jan 2019 - 12:50

score 80 · Answer 10 · 2026-03-25 14:39:52

80

Views

countably compact and compact

Published on 25 Mar 2026 - 14:39

score 441 · Answer 11 · 2026-03-25 22:05:24

441

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 25 Mar 2026 - 22:05

score 277 · Answer 12 · 2026-02-23 04:53:42

277

Views

Equicontinuous homeomorphism and compact metric

Published on 23 Feb 2026 - 4:53

score 715 · Answer 13 · 2026-03-27 08:16:46

715

Views

Compact set always contains its sup and inf

Published on 27 Mar 2026 - 8:16

score 1k · Answer 14 · 2019-01-16 06:48:51

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 16 Jan 2019 - 6:48

score 292 · Answer 15 · 2026-03-29 15:02:27

292

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 29 Mar 2026 - 15:02

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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