Question

Does a sequentially continuous function take its supremum on compacts?

score 64 · Answer 1 · 2019-03-26 15:14:13

64

Views

Does a sequentially continuous function take its supremum on compacts?

Published on 26 Mar 2019 - 15:14

score 48 · Answer 2 · 2019-03-27 18:43:22

48

Views

Continuity of countable projection from non-first countable topological space

Published on 27 Mar 2019 - 18:43

score 163 · Answer 3 · 2019-04-11 16:08:25

Theorem : Every countably compact subset of a first countable Hausdorff space is closed. What happens if the first countability condition is dropped?

score 175 · Answer 4 · 2019-04-29 22:08:26

175

Views

Limit points of sets in a first-countable space have a sequence converging to them

Published on 29 Apr 2019 - 22:08

score 791 · Answer 5 · 2019-05-20 16:58:05

791

Views

Theorem 30.1 (b) in Munkres' TOPOLOGY, 2nd ed: The sequential criterion for continuity

Published on 20 May 2019 - 16:58

score 224 · Answer 6 · 2019-07-22 18:55:41

224

Views

Example of topological space that is not first countable and also is not separable

Published on 22 Jul 2019 - 18:55

score 171 · Answer 7 · 2019-09-21 17:15:54

171

Views

Rudin 4.17. Discontinuous points on a real function can only be at most countable

Published on 21 Sep 2019 - 17:15

score 557 · Answer 8 · 2014-05-02 03:12:43

557

Views

Proofs about continuity and convergence in topological spaces

Published on 02 May 2014 - 3:12

score 511 · Answer 9 · 2014-06-26 22:52:14

511

Views

Prove $\omega_1$ is first countable

Published on 26 Jun 2014 - 22:52

score 2.1k · Answer 10 · 2014-06-29 10:42:29

2.1k

Views

Topological counterexample: compact, Hausdorff, separable space which is not first-countable

Published on 29 Jun 2014 - 10:42

score 1.1k · Answer 11 · 2014-08-23 07:29:05

1.1k

Views

One-point sets are G$_\delta$ in first-countable $T_1$ spaces

Published on 23 Aug 2014 - 7:29

score 1.1k · Answer 12 · 2019-10-08 15:01:31

1.1k

Views

Proof for sequentially continuous function with a first countable domain is continuous

Published on 08 Oct 2019 - 15:01

score 1.2k · Answer 13 · 2019-10-21 17:25:29

1.2k

Views

Is $\mathbb{Q}^n$ countable?

Published on 21 Oct 2019 - 17:25

score 463 · Answer 14 · 2019-11-04 15:22:44

463

Views

Proof that the quotient space of $\mathbb{R}^2/L$ where $L$ is a line passing through the origin is not first countable

Published on 04 Nov 2019 - 15:22

score 394 · Answer 15 · 2019-11-11 03:50:06

394

Views

Cofinite topology if first countable iff $X$ is countable

Published on 11 Nov 2019 - 3:50

List Question

Does a sequentially continuous function take its supremum on compacts?

Continuity of countable projection from non-first countable topological space

Theorem : Every countably compact subset of a first countable Hausdorff space is closed. What happens if the first countability condition is dropped?

Limit points of sets in a first-countable space have a sequence converging to them

Theorem 30.1 (b) in Munkres' TOPOLOGY, 2nd ed: The sequential criterion for continuity

Example of topological space that is not first countable and also is not separable

Rudin 4.17. Discontinuous points on a real function can only be at most countable

Proofs about continuity and convergence in topological spaces

Prove $\omega_1$ is first countable

Topological counterexample: compact, Hausdorff, separable space which is not first-countable

One-point sets are G$_\delta$ in first-countable $T_1$ spaces

Proof for sequentially continuous function with a first countable domain is continuous

Is $\mathbb{Q}^n$ countable?

Proof that the quotient space of $\mathbb{R}^2/L$ where $L$ is a line passing through the origin is not first countable

Cofinite topology if first countable iff $X$ is countable

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