Questions tagged [separable-spaces]

For questions about separable spaces, i.e., topological spaces containing a countable dense set.

A separable space is a topological space which contains a countable dense set.

434 questions
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Why is $L^{\infty}$ not separable?

$l^p (1≤p<{\infty})$ and $L^p (1≤p<∞)$ are separable spaces. What on earth has changed when the value of $p$ turns from a finite number to ${\infty}$? Our teacher gave us some hints that there exists an uncountable subset such that the distance of…
35
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3 answers

Prove if $X$ is a compact metric space, then $X$ is separable.

Let $X$ be a metric space. Prove if $X$ is compact, then $X$ is separable. X separable $\iff X$ contains a countable dense subset. $E \subset X $ dense in $X \iff \overline{E} = X$. $X$ compact $\iff$ every open cover of $X$ admits a finite…
user181728
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32
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1 answer

When is $C_0(X)$ separable?

Recall that a compact Hausdorff space is second countable if and only if the Banach space $C(X)$ of continuous functions on $X$ is separable. I'm looking for a similar criterion for locally compact Hausdorff spaces, using $C_0(X)$ (the Banach space…
30
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5 answers

Why is $l^\infty$ not separable?

My functional analysis textbook says "The metric space $l^\infty$ is not separable." The metric defined between two sequences $\{a_1,a_2,a_3\dots\}$ and $\{b_1,b_2,b_3,\dots\}$ is $\sup\limits_{i\in\Bbb{N}}|{a_i-b_i}|$. How can this be? Isn't the…
28
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2 answers

How to prove that if a normed space has Schauder basis, then it is separable? What about the converse?

Can we take as a dense subset the collection of all the linear combinations of the vectors of the Schauder basis using the rationals as scalars (or the complex numbers with rational real and imaginary parts for that matter)? What can we say about…
28
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1 answer

Can a countable dense subset be split into two disjoint dense subsets?

Suppose $X$ is a compact connected Hausdorff space and $D \subset X$ countable and dense. Can we always write $D=D_1 \cup D_2$ as a disjoint union of countable dense subsets? More generally if $U \subset X$ is open and $D \subset U$ is countable and…
22
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2 answers

Proving separability of the countable product of separable spaces using density.

Let $\{X_j\}_{j=1}^{+\infty}$ be a sequence of separable spaces. The goal is to prove that the infinite cartesian product of separable spaces is indeed separable by showing that the product has a dense subset, arising from the fact that each $X_m$…
21
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2 answers

Prove that $X^\ast$ separable implies $X$ separable

Can someone tell me if I got the following right: Assume $X$ to be a normed vector space over $\mathbb{R}$. Prove that if the dual space $X^\ast$ is separable then $X$ is separable as well. I'm supposed to use the following hint: First show that for…
20
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2 answers

On the product of $\mathfrak c$-many separable spaces

I already figured out how to show that the countable product of separable topological spaces is separable, but I'm out of ideas when the index set has cardinality of $\mathfrak c$. My textbook says it is possible but gives no references. Any…
T. Eskin
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20
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Give an example of a non-separable subspace of a separable space

I'm trying to find an example of a non-separable subspace of a separable space. What kind of examples are there?
19
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2 answers

A metric space is separable iff it is second countable

How do I prove that a metric space is separable iff it is second countable?
16
votes
2 answers

How big can a separable Hausdorff space be?

It is just an idea (might be wrong) but, i think that if a Hausdorff space, say $X$, contains too many elements, then a countable subset cannot be dense in it. Does there exist a cardinality that any space with that (or bigger) cardinality cannot…
ThePortakal
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16
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Proving that a Banach space is separable if its dual is separable

I am currently reading through the proof of the following result: If the dual of a Banach space $X$ is separable, then $X$ is separable. Proof: Let $\{ f_n \}_{n=1}^{\infty}$ be a dense subset of the unit ball $\mathcal{B}$ in $X^{\ast}$. For each…
16
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1 answer

Constraining a dense sequence on a product space, one factor at a time

Slogan: Given a sequence on $X\times Y$, can we choose subsequences to fix the limit in $X$ while leaving the behavior on $Y$ free? Details: Suppose $X$ and $Y$ are topological spaces and $(x_n,y_n)_n$ is a sequence that is "limit-dense" in $X\times…
15
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2 answers

Prove that every compact metric space is separable

I'm in high school and self-studying analysis. I completed this proof for a problem in Rudin, but wanted some verification. Does this look correct? Proof that every compact metric space $K$ has a countable base and is therefore separable: Consider…
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