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 986 · Answer 1 · 2026-03-22 21:40:10

986

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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 22 Mar 2026 - 21:40

score 1.3k · Answer 2 · 2026-03-24 02:16:23

1.3k

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

Published on 24 Mar 2026 - 2:16

score 43 · Answer 3 · 2026-03-25 07:37:57

43

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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 25 Mar 2026 - 7:37

score 29 · Answer 4 · 2026-03-22 19:21:55

29

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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 22 Mar 2026 - 19:21

score 110 · Answer 5 · 2026-03-23 01:12:01

110

Views

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

Published on 23 Mar 2026 - 1:12

score 1.9k · Answer 6 · 2026-03-25 01:37:59

1.9k

Views

A first countable, countably compact space is sequentially compact

Published on 25 Mar 2026 - 1:37

score 578 · Answer 7 · 2026-03-25 02:50:11

578

Views

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

Published on 25 Mar 2026 - 2:50

score 384 · Answer 8 · 2026-03-23 07:14:28

384

Views

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

Published on 23 Mar 2026 - 7:14

score 353 · Answer 9 · 2026-03-22 18:35:24

353

Views

A first countable hemicompact space is locally compact

Published on 22 Mar 2026 - 18:35

score 453 · Answer 10 · 2026-03-22 23:28:16

453

Views

Neighborhood basis in the discrete metric space and first countability

Published on 22 Mar 2026 - 23:28

score 511 · Answer 11 · 2026-03-25 00:31:05

511

Views

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

Published on 25 Mar 2026 - 0:31

score 62 · Answer 12 · 2026-03-25 04:54:38

62

Views

Given set U is first countable or not?

Published on 25 Mar 2026 - 4:54

score 363 · Answer 13 · 2026-03-25 06:24:26

363

Views

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

Published on 25 Mar 2026 - 6:24

score 45 · Answer 14 · 2026-03-25 06:17:50

45

Views

Nested interestection forms neighbourhood basis

Published on 25 Mar 2026 - 6:17

score 64 · Answer 15 · 2026-03-25 06:17:06

64

Views

Countable related question

Published on 25 Mar 2026 - 6:17

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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