Question

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

score 996 · Answer 1 · 2026-05-10 13:22:26

996

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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 10 May 2026 - 13:22

score 1.3k · Answer 2 · 2026-05-10 12:07:00

1.3k

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

Published on 10 May 2026 - 12:07

score 51 · Answer 3 · 2026-04-29 12:55:41

51

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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 36 · Answer 4 · 2026-04-23 19:07:02

36

Views

Let $(X,\tau_x)$ first counable space , $(Y,\tau_y)$, prove that pra image f of an open set in Y, contains open set

Published on 23 Apr 2026 - 19:07

score 117 · Answer 5 · 2026-04-23 07:56:22

117

Views

If a space is first countable, does that imply it is locally compact?

Published on 23 Apr 2026 - 7:56

score 2k · Answer 6 · 2026-04-22 11:03:27

2k

Views

A first countable, countably compact space is sequentially compact

Published on 22 Apr 2026 - 11:03

score 583 · Answer 7 · 2026-04-22 17:51:56

583

Views

The continuous image of a First Countable Space need not be First Countable (Willard 16.B.1)

Published on 22 Apr 2026 - 17:51

score 391 · Answer 8 · 2026-04-23 23:09:33

391

Views

Is the topology of weak convergence of probability measures first-countable?

Published on 23 Apr 2026 - 23:09

score 358 · Answer 9 · 2026-04-23 00:06:39

358

Views

A first countable hemicompact space is locally compact

Published on 23 Apr 2026 - 0:06

score 461 · Answer 10 · 2026-04-23 07:47:49

461

Views

Neighborhood basis in the discrete metric space and first countability

Published on 23 Apr 2026 - 7:47

score 518 · Answer 11 · 2026-04-30 23:34:46

518

Views

What is the proof that first countable is sufficient to say that sequentially closed implies closed

Published on 30 Apr 2026 - 23:34

score 65 · Answer 12 · 2026-04-28 12:06:59

65

Views

Given set U is first countable or not?

Published on 28 Apr 2026 - 12:06

score 366 · Answer 13 · 2026-04-28 08:21:55

366

Views

Does first countable imply equivalence of sequential and limit point compactness?

Published on 28 Apr 2026 - 8:21

score 50 · Answer 14 · 2026-05-06 09:22:59

50

Views

Nested interestection forms neighbourhood basis

Published on 06 May 2026 - 9:22

score 70 · Answer 15 · 2026-04-30 09:29:39

70

Views

Countable related question

Published on 30 Apr 2026 - 9:29

List Question

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.

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

Let $(X,\tau_x)$ first counable space , $(Y,\tau_y)$, prove that pra image f of an open set in Y, contains open set

If a space is first countable, does that imply it is locally compact?

A first countable, countably compact space is sequentially compact

The continuous image of a First Countable Space need not be First Countable (Willard 16.B.1)

Is the topology of weak convergence of probability measures first-countable?

A first countable hemicompact space is locally compact

Neighborhood basis in the discrete metric space and first countability

What is the proof that first countable is sufficient to say that sequentially closed implies closed

Given set U is first countable or not?

Does first countable imply equivalence of sequential and limit point compactness?

Nested interestection forms neighbourhood basis

Countable related question

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