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.

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

votes

$l^p (1≤p<{\infty})$ 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…

Andylang

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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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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…

Paul Siegel

- 8,597
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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…

user67803

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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…

Saaqib Mahmood

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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…

Daron

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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 $X_m$…

Moe

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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…

Rudy the Reindeer

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

nomadicmathematician

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How do I prove that a metric space is separable iff it is second countable?

Heisenberg

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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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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…

Shaun Davis

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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…

Chris Culter

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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…

lithium123

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