Question

Does a sequentially continuous function take its supremum on compacts?

score 67 · Answer 1 · 2026-04-10 11:35:13

67

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Does a sequentially continuous function take its supremum on compacts?

Published on 10 Apr 2026 - 11:35

score 51 · Answer 2 · 2026-04-12 22:48:51

51

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Continuity of countable projection from non-first countable topological space

Published on 12 Apr 2026 - 22:48

score 168 · Answer 3 · 2026-04-16 23:42:22

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

score 178 · Answer 4 · 2026-04-14 14:57:16

178

Views

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

Published on 14 Apr 2026 - 14:57

score 794 · Answer 5 · 2026-04-14 15:18:46

794

Views

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

Published on 14 Apr 2026 - 15:18

score 227 · Answer 6 · 2026-04-11 07:21:21

227

Views

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

Published on 11 Apr 2026 - 7:21

score 174 · Answer 7 · 2026-04-16 14:54:39

174

Views

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

Published on 16 Apr 2026 - 14:54

score 560 · Answer 8 · 2026-04-11 02:31:40

560

Views

Proofs about continuity and convergence in topological spaces

Published on 11 Apr 2026 - 2:31

score 515 · Answer 9 · 2026-04-09 09:18:34

515

Views

Prove $\omega_1$ is first countable

Published on 09 Apr 2026 - 9:18

score 2.1k · Answer 10 · 2026-04-09 07:11:34

2.1k

Views

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

Published on 09 Apr 2026 - 7:11

score 1.1k · Answer 11 · 2026-04-13 21:12:20

1.1k

Views

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

Published on 13 Apr 2026 - 21:12

score 1.1k · Answer 12 · 2026-05-06 16:11:25

1.1k

Views

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

Published on 06 May 2026 - 16:11

score 1.2k · Answer 13 · 2026-04-01 12:45:55

1.2k

Views

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

Published on 01 Apr 2026 - 12:45

score 464 · Answer 14 · 2026-04-06 13:31:41

464

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 06 Apr 2026 - 13:31

score 396 · Answer 15 · 2026-04-13 07:14:06

396

Views

Cofinite topology if first countable iff $X$ is countable

Published on 13 Apr 2026 - 7:14

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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