Question

Every subspace of a separable metric space is separable

score 245 · Answer 1 · 2020-02-05 22:37:38

245

Views

Every subspace of a separable metric space is separable

Published on 05 Feb 2020 - 22:37

score 207 · Answer 2 · 2026-03-19 15:02:04

207

Views

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

Published on 19 Mar 2026 - 15:02

score 40 · Answer 3 · 2020-02-21 06:01:21

40

Views

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

Published on 21 Feb 2020 - 6:01

score 62 · Answer 4 · 2020-03-06 23:42:46

62

Views

Separable vs countable

Published on 06 Mar 2020 - 23:42

score 76 · Answer 5 · 2026-03-19 12:28:10

76

Views

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

Published on 19 Mar 2026 - 12:28

score 47 · Answer 6 · 2026-03-23 21:57:58

47

Views

About countable set

Published on 23 Mar 2026 - 21:57

score 76 · Answer 7 · 2020-03-24 14:25:26

76

Views

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

Published on 24 Mar 2020 - 14:25

score 239 · Answer 8 · 2020-04-20 14:11:14

239

Views

Radially open set topology is separable?

Published on 20 Apr 2020 - 14:11

score 130 · Answer 9 · 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

score 393 · Answer 10 · 2020-04-27 14:56:13

393

Views

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

Published on 27 Apr 2020 - 14:56

score 192 · Answer 11 · 2020-04-27 19:32:01

192

Views

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

Published on 27 Apr 2020 - 19:32

score 346 · Answer 12 · 2026-02-23 02:59:31

346

Views

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

Published on 23 Feb 2026 - 2:59

score 74 · Answer 13 · 2020-04-29 10:14:11

74

Views

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

Published on 29 Apr 2020 - 10:14

score 164 · Answer 14 · 2020-04-30 21:57:02

164

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 30 Apr 2020 - 21:57

score 145 · Answer 15 · 2020-05-02 21:10:29

145

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 02 May 2020 - 21:10

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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