Question

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

score 291 · Answer 1 · 2026-05-06 07:33:54

291

Views

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

Published on 06 May 2026 - 7:33

score 995 · Answer 2 · 2026-05-06 10:03:34

995

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 06 May 2026 - 10:03

score 1.3k · Answer 3 · 2026-04-29 12:52:16

1.3k

Views

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

Published on 29 Apr 2026 - 12:52

score 297 · Answer 4 · 2026-05-02 23:46:39

297

Views

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

Published on 02 May 2026 - 23:46

score 159 · Answer 5 · 2026-04-25 08:38:06

159

Views

Countable union covers a second countable, compact Hausdorff space

Published on 25 Apr 2026 - 8:38

score 2.2k · Answer 6 · 2026-04-29 12:51:49

2.2k

Views

Compactness gives Second countable space??

Published on 29 Apr 2026 - 12:51

score 527 · Answer 7 · 2026-05-06 10:03:25

527

Views

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

Published on 06 May 2026 - 10:03

score 1.9k · Answer 8 · 2026-04-29 12:52:20

1.9k

Views

A Countable Basis and Countably Locally Finite Collections

Published on 29 Apr 2026 - 12:52

score 895 · Answer 9 · 2026-05-02 05:16:49

895

Views

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

Published on 02 May 2026 - 5:16

score 194 · Answer 10 · 2026-04-29 12:51:43

194

Views

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

Published on 29 Apr 2026 - 12:51

score 1k · Answer 11 · 2026-05-05 16:22:06

1k

Views

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

Published on 05 May 2026 - 16:22

score 657 · Answer 12 · 2026-04-23 18:54:39

657

Views

Does partition of unity implies second countable?

Published on 23 Apr 2026 - 18:54

score 51 · Answer 13 · 2026-04-29 12:55:41

51

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 29 Apr 2026 - 12:55

score 266 · Answer 14 · 2026-04-29 12:55:42

266

Views

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

Published on 29 Apr 2026 - 12:55

score 78 · Answer 15 · 2026-04-27 16:02:36

78

Views

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

Published on 27 Apr 2026 - 16:02

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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