Question

$X$ compact Hausdorff space whose diagonal is a zero set: prove that the diagonal in $X \times X$ is a $G_{\delta}$ -set

score 249 · Answer 1 · 2026-03-25 20:15:52

249

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$X$ compact Hausdorff space whose diagonal is a zero set: prove that the diagonal in $X \times X$ is a $G_{\delta}$ -set

Published on 25 Mar 2026 - 20:15

score 395 · Answer 2 · 2026-03-25 20:09:26

395

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Separable, metrizable, and second countable spaces

Published on 25 Mar 2026 - 20:09

score 65 · Answer 3 · 2026-04-09 07:48:48

65

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Prove that $\mathbb R_l$ $\times$ $\mathbb R_l$ given the box topology has a countable dense subset.

Published on 09 Apr 2026 - 7:48

score 275 · Answer 4 · 2026-03-25 20:12:01

275

Views

Give a subspace of $\mathbb R_l \times \mathbb R_l$ that has no countable dense subset [relative to the subspace topology].

Published on 25 Mar 2026 - 20:12

score 746 · Answer 5 · 2026-03-25 20:12:57

746

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Indiscrete topological space is second countable.

Published on 25 Mar 2026 - 20:12

score 1.2k · Answer 6 · 2026-04-12 18:26:41

1.2k

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Determine if $\mathbb{R}^{\omega}$ with the product topology is second countable

Published on 12 Apr 2026 - 18:26

score 1.7k · Answer 7 · 2026-03-25 20:11:58

1.7k

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Prove that $\mathbb{R}_l$ is not a second countable. ($\mathbb{R}_l$ are the real ones with the topology of the lower limit)

Published on 25 Mar 2026 - 20:11

score 108 · Answer 8 · 2026-03-25 20:11:55

108

Views

How does the underlying topological space of manifold depend on the properties of the atlas?

Published on 25 Mar 2026 - 20:11

score 176 · Answer 9 · 2026-03-25 20:09:50

176

Views

$X^$ with $w^$-topology second countable for separable Banach space $X$?

Published on 25 Mar 2026 - 20:09

score 1.4k · Answer 10 · 2026-04-12 03:06:52

1.4k

Views

Does there exist a Connected Locally Euclidean Space that is not second countable?

Published on 12 Apr 2026 - 3:06

score 4.5k · Answer 11 · 2026-04-12 09:26:28

4.5k

Views

Hausdorff locally compact and second countable is $\sigma$-compact

Published on 12 Apr 2026 - 9:26

score 357 · Answer 12 · 2026-04-17 12:53:51

357

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In this proof that a compact metric space is second countable, why exactly is the induced topology the same as the metric topology?

Published on 17 Apr 2026 - 12:53

score 1.8k · Answer 13 · 2026-04-11 01:09:30

1.8k

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Does proving (second countable) $\Rightarrow$ (Lindelöf) require the axiom of choice?

Published on 11 Apr 2026 - 1:09

score 2.3k · Answer 14 · 2026-04-10 17:45:32

2.3k

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Equivalence of three properties of a metric space.

Published on 10 Apr 2026 - 17:45

score 737 · Answer 15 · 2026-04-12 21:18:35

737

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Being second countable is invariant under perfect mapping

Published on 12 Apr 2026 - 21:18

List Question

$X$ compact Hausdorff space whose diagonal is a zero set: prove that the diagonal in $X \times X$ is a $G_{\delta}$ -set

Separable, metrizable, and second countable spaces

Prove that $\mathbb R_l$ $\times$ $\mathbb R_l$ given the box topology has a countable dense subset.

Give a subspace of $\mathbb R_l \times \mathbb R_l$ that has no countable dense subset [relative to the subspace topology].

Indiscrete topological space is second countable.

Determine if $\mathbb{R}^{\omega}$ with the product topology is second countable

Prove that $\mathbb{R}_l$ is not a second countable. ($\mathbb{R}_l$ are the real ones with the topology of the lower limit)

How does the underlying topological space of manifold depend on the properties of the atlas?

$X^$ with $w^$-topology second countable for separable Banach space $X$?

Does there exist a Connected Locally Euclidean Space that is not second countable?

Hausdorff locally compact and second countable is $\sigma$-compact

In this proof that a compact metric space is second countable, why exactly is the induced topology the same as the metric topology?

Does proving (second countable) $\Rightarrow$ (Lindelöf) require the axiom of choice?

Equivalence of three properties of a metric space.

Being second countable is invariant under perfect mapping

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