Question

Showing that a first countable space is Hausdorff if and only if every sequence converging in $X$ has a unique limit

score 1.1k · Answer 1 · 2019-11-17 20:06:10

1.1k

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Showing that a first countable space is Hausdorff if and only if every sequence converging in $X$ has a unique limit

Published on 17 Nov 2019 - 20:06

score 85 · Answer 2 · 2026-03-07 03:39:41

85

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Prove that the set of all differentiable functions :[0,1]→ [0,1] is uncountable.

Published on 07 Mar 2026 - 3:39

score 252 · Answer 3 · 2026-04-30 05:40:12

252

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First countable space

Published on 30 Apr 2026 - 5:40

score 182 · Answer 4 · 2026-03-25 09:44:41

182

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Continuous open maps and first countability

Published on 25 Mar 2026 - 9:44

score 302 · Answer 5 · 2026-03-25 09:53:39

302

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Locally euclidean and first countability

Published on 25 Mar 2026 - 9:53

score 96 · Answer 6 · 2026-03-25 09:50:04

96

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Can $X \setminus Y$ be first category? For $Y$ a proper linear subspace of a completely metrizable TVS $X$

Published on 25 Mar 2026 - 9:50

score 70 · Answer 7 · 2026-03-25 09:50:10

70

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Baby Rudin 2.12 vs power set of N

Published on 25 Mar 2026 - 9:50

score 138 · Answer 8 · 2026-03-25 09:48:57

138

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I want to prove that $\mathbb{Q}^{\infty}$ is not first countable.

Published on 25 Mar 2026 - 9:48

score 64 · Answer 9 · 2026-05-10 16:50:55

64

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About countable set

Published on 10 May 2026 - 16:50

score 205 · Answer 10 · 2026-03-25 09:48:33

205

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Conclude that the following topological vector space is not first-countable

Published on 25 Mar 2026 - 9:48

score 241 · Answer 11 · 2026-05-10 15:44:34

241

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Radially open set topology is separable?

Published on 10 May 2026 - 15:44

score 50 · Answer 12 · 2026-03-25 11:25:59

50

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In which of the three topologies does $X$ have a countable basis?

Published on 25 Mar 2026 - 11:25

score 193 · Answer 13 · 2026-03-25 09:50:06

193

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Prob. 1 (b), Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Example of a not-first-countable space with every singleton set being a $G_\delta$ set

Published on 25 Mar 2026 - 9:50

score 166 · Answer 14 · 2026-05-10 15:47:18

166

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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 10 May 2026 - 15:47

score 47 · Answer 15 · 2026-03-25 09:53:30

47

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Showing that no point of $X^*$ is a point at which $\beta X$ is first countable.

Published on 25 Mar 2026 - 9:53

List Question

Showing that a first countable space is Hausdorff if and only if every sequence converging in $X$ has a unique limit

Prove that the set of all differentiable functions :[0,1]→ [0,1] is uncountable.

First countable space

Continuous open maps and first countability

Locally euclidean and first countability

Can $X \setminus Y$ be first category? For $Y$ a proper linear subspace of a completely metrizable TVS $X$

Baby Rudin 2.12 vs power set of N

I want to prove that $\mathbb{Q}^{\infty}$ is not first countable.

About countable set

Conclude that the following topological vector space is not first-countable

Radially open set topology is separable?

In which of the three topologies does $X$ have a countable basis?

Prob. 1 (b), Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Example of a not-first-countable space with every singleton set being a $G_\delta$ set

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

Showing that no point of $X^*$ is a point at which $\beta X$ is first countable.

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