Question

Space of holomorphic functions with compact-open topology is second-countable

score 280 · Answer 1 · 2026-03-22 23:30:26

280

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Space of holomorphic functions with compact-open topology is second-countable

Published on 22 Mar 2026 - 23:30

score 986 · Answer 2 · 2026-03-22 21:40:10

986

Views

Prove that $\prod X_\alpha$ is first (second)-countable if and only if $X_\alpha$ is first (second)-countable, $\forall \alpha\in I.$

Published on 22 Mar 2026 - 21:40

score 1.3k · Answer 3 · 2026-03-24 02:16:23

1.3k

Views

If X satisfies the first or second countability axiom then $F(X)$ satisfies the same condition.

Published on 24 Mar 2026 - 2:16

score 289 · Answer 4 · 2026-03-20 05:59:40

289

Views

Unit ball of the adjoint space of a separable Banach space is second-countable in the weak* topology.

Published on 20 Mar 2026 - 5:59

score 151 · Answer 5 · 2026-03-24 06:42:59

151

Views

Countable union covers a second countable, compact Hausdorff space

Published on 24 Mar 2026 - 6:42

score 2.2k · Answer 6 · 2026-03-22 16:41:43

2.2k

Views

Compactness gives Second countable space??

Published on 22 Mar 2026 - 16:41

score 516 · Answer 7 · 2026-03-22 19:33:42

516

Views

Show that if a topological space has at most countable basis, then the space is separable and Lindelöf

Published on 22 Mar 2026 - 19:33

score 1.9k · Answer 8 · 2026-03-22 23:00:57

1.9k

Views

A Countable Basis and Countably Locally Finite Collections

Published on 22 Mar 2026 - 23:00

score 883 · Answer 9 · 2026-03-22 17:33:06

883

Views

$\Bbb{R}^\omega$ with the Uniform Metric is not 2nd Countable

Published on 22 Mar 2026 - 17:33

score 187 · Answer 10 · 2026-03-23 09:42:10

187

Views

Compact T2 space with separable C(X) is second-countable

Published on 23 Mar 2026 - 9:42

score 1k · Answer 11 · 2026-03-24 21:16:06

1k

Views

A space is metrisable if and only if it admits a countable basis

Published on 24 Mar 2026 - 21:16

score 642 · Answer 12 · 2026-03-22 14:19:47

642

Views

Does partition of unity implies second countable?

Published on 22 Mar 2026 - 14:19

score 42 · Answer 13 · 2026-03-17 01:22:56

42

Views

Check properties of the topology $\{U \subseteq \mathbb{R} \mid \forall x \in U: \exists \epsilon > 0: [x, x + \epsilon[ \subseteq U\}$

Published on 17 Mar 2026 - 1:22

score 258 · Answer 14 · 2026-03-22 15:20:13

258

Views

$\mathbb{R}_{\ell}$ is not second countable

Published on 22 Mar 2026 - 15:20

score 72 · Answer 15 · 2026-03-17 16:03:50

72

Views

$\coprod_{i \in I} X_i$ is second countable $\implies |I| \leq |\mathbb{N}|$

Published on 17 Mar 2026 - 16:03

List Question

Space of holomorphic functions with compact-open topology is second-countable

Prove that $\prod X_\alpha$ is first (second)-countable if and only if $X_\alpha$ is first (second)-countable, $\forall \alpha\in I.$

If X satisfies the first or second countability axiom then $F(X)$ satisfies the same condition.

Unit ball of the adjoint space of a separable Banach space is second-countable in the weak* topology.

Countable union covers a second countable, compact Hausdorff space

Compactness gives Second countable space??

Show that if a topological space has at most countable basis, then the space is separable and Lindelöf

A Countable Basis and Countably Locally Finite Collections

$\Bbb{R}^\omega$ with the Uniform Metric is not 2nd Countable

Compact T2 space with separable C(X) is second-countable

A space is metrisable if and only if it admits a countable basis

Does partition of unity implies second countable?

Check properties of the topology $\{U \subseteq \mathbb{R} \mid \forall x \in U: \exists \epsilon > 0: [x, x + \epsilon[ \subseteq U\}$

$\mathbb{R}_{\ell}$ is not second countable

$\coprod_{i \in I} X_i$ is second countable $\implies |I| \leq |\mathbb{N}|$

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