Question

Let $x\in X$ and $S\subset X$ be countable and dense. Is there a sequence in $S$ converging to $x$?

score 43 · Answer 1 · 2017-02-06 15:05:16

43

Views

Let $x\in X$ and $S\subset X$ be countable and dense. Is there a sequence in $S$ converging to $x$?

Published on 06 Feb 2017 - 15:05

score 1.3k · Answer 2 · 2017-02-08 14:22:33

1.3k

Views

Normed space $V$ separable $\Leftrightarrow$ unit sphere $\{x\in V: |x|=1\}$ separable

Published on 08 Feb 2017 - 14:22

score 101 · Answer 3 · 2017-02-16 10:28:35

101

Views

Separability of product metric spaces

Published on 16 Feb 2017 - 10:28

score 89 · Answer 4 · 2017-02-20 01:48:46

89

Views

If a set is bigger than a dense set in cardinality, does it have a limit point?

Published on 20 Feb 2017 - 1:48

score 127 · Answer 5 · 2026-03-25 15:42:29

127

Views

$X$ Banach space , $T \in B(X , l^ {\infty})$ be surjective , there is a bounded sequence $\{x_n\}$ in $X$ such that $T(x_n)=e_n$?

Published on 25 Mar 2026 - 15:42

score 572 · Answer 6 · 2017-02-22 17:55:33

572

Views

Separability of the space of sequences of real numbers $R^\omega$ with the uniform topology

Published on 22 Feb 2017 - 17:55

score 129 · Answer 7 · 2026-03-25 13:51:37

129

Views

second-countability of the upper halfplane

Published on 25 Mar 2026 - 13:51

score 95 · Answer 8 · 2017-02-28 08:06:15

95

Views

$X,Y$ be metric spaces , $f:X \to Y$ be continuous and injective ; if $f(X)$ is separable , then is $X$ also separable?

Published on 28 Feb 2017 - 8:06

score 564 · Answer 9 · 2017-02-28 12:58:38

564

Views

Can every separable Banach space be isometrically embedded in $l^2$ ? Or at least in $l^p$ for some $1\le p<\infty$?

Published on 28 Feb 2017 - 12:58

score 322 · Answer 10 · 2017-03-03 19:23:09

322

Views

Proof [0,1]^[0,1] is separable

Published on 03 Mar 2017 - 19:23

score 84 · Answer 11 · 2017-03-05 12:49:06

84

Views

Is every $T_1$ topological space homeomorphic to a subspace of a separable $T_1$ topological space (with the same cardinality)?

Published on 05 Mar 2017 - 12:49

score 167 · Answer 12 · 2017-03-26 17:03:41

167

Views

Proving that the space with norm; $\left\|x\right\|_p=\left(\sum_{i=1}^{n}\left|x_i\right|^p\right)^{1/p}$ is separable

Published on 26 Mar 2017 - 17:03

score 444 · Answer 13 · 2017-03-27 04:26:48

444

Views

Proving that the closed unit sphere of a separable normed space is separable

Published on 27 Mar 2017 - 4:26

score 168 · Answer 14 · 2026-03-25 08:09:57

168

Views

Existance of Schauder Basis of vector space over field F implies separability

Published on 25 Mar 2026 - 8:09

score 157 · Answer 15 · 2023-02-05 22:35:27

157

Views

Is it possible to define a topology on the real line such that 0 and non-zero integers are dense but no finite subset of non-zero integers is dense?

Published on 05 Feb 2023 - 22:35

List Question

Let $x\in X$ and $S\subset X$ be countable and dense. Is there a sequence in $S$ converging to $x$?

Normed space $V$ separable $\Leftrightarrow$ unit sphere $\{x\in V: |x|=1\}$ separable

Separability of product metric spaces

If a set is bigger than a dense set in cardinality, does it have a limit point?

$X$ Banach space , $T \in B(X , l^ {\infty})$ be surjective , there is a bounded sequence $\{x_n\}$ in $X$ such that $T(x_n)=e_n$?

Separability of the space of sequences of real numbers $R^\omega$ with the uniform topology

second-countability of the upper halfplane

$X,Y$ be metric spaces , $f:X \to Y$ be continuous and injective ; if $f(X)$ is separable , then is $X$ also separable?

Can every separable Banach space be isometrically embedded in $l^2$ ? Or at least in $l^p$ for some $1\le p<\infty$?

Proof [0,1]^[0,1] is separable

Is every $T_1$ topological space homeomorphic to a subspace of a separable $T_1$ topological space (with the same cardinality)?

Proving that the space with norm; $\left\|x\right\|_p=\left(\sum_{i=1}^{n}\left|x_i\right|^p\right)^{1/p}$ is separable

Proving that the closed unit sphere of a separable normed space is separable

Existance of Schauder Basis of vector space over field F implies separability

Is it possible to define a topology on the real line such that 0 and non-zero integers are dense but no finite subset of non-zero integers is dense?

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