List questions from compactness

Published on 11 Jul 2018 - 18:42

443

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Clean Proof that a set is bounded in multiple dimension

Published on 25 Mar 2026 - 23:16

341

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"Decreasing" sequence of nonempty compact sets has a nonempty intersection.

Published on 12 Jul 2018 - 10:52

73

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Sequentially compact

Published on 12 Jul 2018 - 11:43

2.3k

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Every sequence has a weakly convergent subsequence with limit

Published on 23 Mar 2026 - 0:36

338

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Rudin's theorem on compact sets and limit points

Published on 12 Jul 2018 - 20:47

88

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Show we can arrange a $\frac{\delta}{2}$-net of a compact and connected metric space X such that distance between subsequent points is $\lt \delta$.

Published on 12 Jul 2018 - 21:36

90

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Try to find a contradiction in showing a unit ball is not weakly compact.

Published on 28 Mar 2026 - 10:16

544

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Rudin's proof on a nested sequence of k-cells

Published on 13 Jul 2018 - 1:20

216

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If $X$ is a compact subset of $M$, $y \in X'$, then there exists a point $a \in X$ such that $d(a,y) \leq d(x,y)$ for all $x \in X$

Published on 13 Jul 2018 - 15:47

514

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Munkres proof that $ \mathbb{R}^{ \omega } $ is not locally compact (clarification needed)

Published on 14 Jul 2018 - 3:19

345

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Regarding a metric space is compact, complete, connected

Published on 25 Mar 2026 - 15:10

54

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$X$ is c-closed iff every countably compact subset of $X$ is closed

Published on 14 Jul 2018 - 22:27

235

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If $C$ is a compact set and $(a,b)$ is any open interval, then $C\setminus (a,b)$ is a compact set.

Published on 15 Jul 2018 - 15:32

143

Views

Proof of Tychonoff's Theorem using subbasic open subsets

Published on 25 Mar 2026 - 9:24

List Question

One Critical Value despite being on a compact set

Clean Proof that a set is bounded in multiple dimension

"Decreasing" sequence of nonempty compact sets has a nonempty intersection.

Sequentially compact

Every sequence has a weakly convergent subsequence with limit

Rudin's theorem on compact sets and limit points

Show we can arrange a $\frac{\delta}{2}$-net of a compact and connected metric space X such that distance between subsequent points is $\lt \delta$.

Try to find a contradiction in showing a unit ball is not weakly compact.

Rudin's proof on a nested sequence of k-cells

If $X$ is a compact subset of $M$, $y \in X'$, then there exists a point $a \in X$ such that $d(a,y) \leq d(x,y)$ for all $x \in X$

Munkres proof that $ \mathbb{R}^{ \omega } $ is not locally compact (clarification needed)

Regarding a metric space is compact, complete, connected

$X$ is c-closed iff every countably compact subset of $X$ is closed

If $C$ is a compact set and $(a,b)$ is any open interval, then $C\setminus (a,b)$ is a compact set.

Proof of Tychonoff's Theorem using subbasic open subsets

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