Question

Prove that if $\forall n \neq m$, we have $d(x_n, x_m) = 1$, then no subsequence of $x_n$ converges.

score 56 · Answer 1 · 2020-04-17 01:16:36

56

Views

Prove that if $\forall n \neq m$, we have $d(x_n, x_m) = 1$, then no subsequence of $x_n$ converges.

Published on 17 Apr 2020 - 1:16

score 548 · Answer 2 · 2026-03-25 19:04:27

548

Views

Nets, Cluster points, and closure: Showing a space is compact if every net has a cluster point

Published on 25 Mar 2026 - 19:04

score 83 · Answer 3 · 2026-03-26 14:21:09

83

Views

A counterpart of Bolzano-Weierstass theorem in metric space.

Published on 26 Mar 2026 - 14:21

score 123 · Answer 4 · 2020-04-17 08:21:51

123

Views

Proving that the infinite intersection of a sequence of closed subsets in a compact space is non-empty.

Published on 17 Apr 2020 - 8:21

score 40 · Answer 5 · 2020-04-17 16:21:30

40

Views

Disk compactness

Published on 17 Apr 2020 - 16:21

score 271 · Answer 6 · 2020-04-17 17:14:24

271

Views

Is every compact discrete metric space singleton?

Published on 17 Apr 2020 - 17:14

score 314 · Answer 7 · 2020-04-18 03:49:22

314

Views

Compact Sets Have Finite Diameter -- Trouble with Proof

Published on 18 Apr 2020 - 3:49

score 171 · Answer 8 · 2026-04-01 00:39:22

171

Views

Separating a point and a compact subset in a Hausdorff space without Choice

Published on 01 Apr 2026 - 0:39

score 121 · Answer 9 · 2026-03-29 03:27:39

121

Views

Isolated points and Baire category theorem

Published on 29 Mar 2026 - 3:27

score 22 · Answer 10 · 2020-04-20 06:14:16

22

Views

Role of $K$ be a compact space in showing that $L$ is positive.

Published on 20 Apr 2020 - 6:14

score 81 · Answer 11 · 2020-04-20 13:12:27

81

Views

Prove this $f$ has an absolute maximum.

Published on 20 Apr 2020 - 13:12

score 67 · Answer 12 · 2020-04-20 15:49:57

67

Views

Prove that monotone sequence in $K$ that does not converges to a point in $K$

Published on 20 Apr 2020 - 15:49

score 267 · Answer 13 · 2020-04-21 19:47:19

267

Views

Proving that the set of real orthogonal $n \times n$ matrices is compact in $ M^{n \times n}( \mathbb{R})$.

Published on 21 Apr 2020 - 19:47

score 173 · Answer 14 · 2026-03-29 16:55:57

173

Views

Group action a Compact Hausdorff space

Published on 29 Mar 2026 - 16:55

score 130 · Answer 15 · 2026-02-24 05:36:19

130

Views

Prove the compactness and second countability of a subset of $\Bbb{R}^{L^1(G)\times D}$

Published on 24 Feb 2026 - 5:36

List Question

Prove that if $\forall n \neq m$, we have $d(x_n, x_m) = 1$, then no subsequence of $x_n$ converges.

Nets, Cluster points, and closure: Showing a space is compact if every net has a cluster point

A counterpart of Bolzano-Weierstass theorem in metric space.

Proving that the infinite intersection of a sequence of closed subsets in a compact space is non-empty.

Disk compactness

Is every compact discrete metric space singleton?

Compact Sets Have Finite Diameter -- Trouble with Proof

Separating a point and a compact subset in a Hausdorff space without Choice

Isolated points and Baire category theorem

Role of $K$ be a compact space in showing that $L$ is positive.

Prove this $f$ has an absolute maximum.

Prove that monotone sequence in $K$ that does not converges to a point in $K$

Proving that the set of real orthogonal $n \times n$ matrices is compact in $ M^{n \times n}( \mathbb{R})$.

Group action a Compact Hausdorff space

Prove the compactness and second countability of a subset of $\Bbb{R}^{L^1(G)\times D}$

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