Question

Every subspace of a separable metric space is separable

score 246 · Answer 1 · 2026-03-25 10:53:02

246

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Every subspace of a separable metric space is separable

Published on 25 Mar 2026 - 10:53

score 217 · Answer 2 · 2026-04-28 18:34:41

217

Views

Second Countability $\implies$ Lindelöf property(Proof verification)

Published on 28 Apr 2026 - 18:34

score 41 · Answer 3 · 2026-03-25 10:55:38

41

Views

Is the set $A=\{x\in l_p :x_n \in \mathbb Q\}$ countable?

Published on 25 Mar 2026 - 10:55

score 64 · Answer 4 · 2026-04-05 16:32:03

64

Views

Separable vs countable

Published on 05 Apr 2026 - 16:32

score 83 · Answer 5 · 2026-04-30 00:40:36

83

Views

$Y\subset X$ is Lindelof iff Every $X$-open cover of $Y$ has a countable $X$-open subcover.

Published on 30 Apr 2026 - 0:40

score 63 · Answer 6 · 2026-04-30 06:04:43

63

Views

About countable set

Published on 30 Apr 2026 - 6:04

score 77 · Answer 7 · 2026-03-25 12:28:26

77

Views

Proving a well known result of metric spaces without using Zorn's Lemma.

Published on 25 Mar 2026 - 12:28

score 240 · Answer 8 · 2026-03-25 09:46:58

240

Views

Radially open set topology is separable?

Published on 25 Mar 2026 - 9:46

score 131 · Answer 9 · 2026-04-12 08:28:23

131

Views

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

Published on 12 Apr 2026 - 8:28

score 394 · Answer 10 · 2026-03-25 10:57:07

394

Views

Prob. 2, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Every basis of a second-countable space contains a countable basis

Published on 25 Mar 2026 - 10:57

score 193 · Answer 11 · 2026-03-25 10:55:51

193

Views

Prob. 4, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Every compact metrizable space has a countable basis.

Published on 25 Mar 2026 - 10:55

score 350 · Answer 12 · 2026-05-01 15:00:21

350

Views

Prob. 6, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: $\mathbb{R}_\ell$ and $I_0^2$ are not metrizable

Published on 01 May 2026 - 15:00

score 77 · Answer 13 · 2026-04-14 23:28:50

77

Views

Is the product space of a non-second-countable space and a second-countable space non-second-countable?

Published on 14 Apr 2026 - 23:28

score 165 · Answer 14 · 2026-03-25 09:49:28

165

Views

Prob. 12, Sec. 30, in Munkres' TOPOLOGY, 2nd edition: Every continuous open image of a first / second countable space is also first / second countable

Published on 25 Mar 2026 - 9:49

score 146 · Answer 15 · 2026-03-25 12:28:07

146

Views

Prob. 15, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: The subspace $\mathscr{C}(I,\mathbb{R})$ of $\mathbb{R}^I$ with uniform metric is separable

Published on 25 Mar 2026 - 12:28

List Question

Every subspace of a separable metric space is separable

Second Countability $\implies$ Lindelöf property(Proof verification)

Is the set $A=\{x\in l_p :x_n \in \mathbb Q\}$ countable?

Separable vs countable

$Y\subset X$ is Lindelof iff Every $X$-open cover of $Y$ has a countable $X$-open subcover.

About countable set

Proving a well known result of metric spaces without using Zorn's Lemma.

Radially open set topology is separable?

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

Prob. 2, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Every basis of a second-countable space contains a countable basis

Prob. 4, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Every compact metrizable space has a countable basis.

Prob. 6, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: $\mathbb{R}_\ell$ and $I_0^2$ are not metrizable

Is the product space of a non-second-countable space and a second-countable space non-second-countable?

Prob. 12, Sec. 30, in Munkres' TOPOLOGY, 2nd edition: Every continuous open image of a first / second countable space is also first / second countable

Prob. 15, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: The subspace $\mathscr{C}(I,\mathbb{R})$ of $\mathbb{R}^I$ with uniform metric is separable

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