# Questions tagged [separable-spaces]

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

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### $X^*$ separable implies separable unit ball

Let $X$ be the dual space of a normed space $X$ and let $X^*$ be separable. Prove that $S^* = \left\{F \in X^* : ||F|| = 1\right\}$ is separable. My solution : suppose $Y$ is a dense countable set in ...
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### Is this Proof of if $X$ be a separable normed linear space then its closed unit sphere is also separable correct?

Here is the link to the proof: Proving that normed space is separable iff its unit sphere is separable And here is the proof: Yes, it holds $\partial S = S$. We have \partial S = \overline{S} \...
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### Sublagebra of Right uniform continuous functions on a separable, metrisable groupG with countable dense subset D is G invariant iff D invariant

Motivation/backrgound: I am reading about some results in topological dynamics, and particularly concerning the action of seperable metrisable groups G acting on compact Hausdorff spaces. I am trying ...
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### Converse statements regarding separable metric space

These statements: a. Every infinite subset has a limit point b. Separable c. Has a countable base d. Compact are mentioned in the exercises of baby Rudin Chapter 2 I have proven with hints (in ...
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### Is the set $A=\{x\in l_p :x_n \in \mathbb Q\}$ countable? [closed]

Is the set $A=\{x\in \it l_p :x_n \in \mathbb Q\}$ countable?If yes,then I can show that $l_p$ space is separable with resepect to the metric $d(x,y)=(\sum _nx_n^p)^{1/p}$.
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### Closure of smooth compactly supported functions separable

Consider the norm $\|f\|=\sup_{x>0}|xf^\prime(x)|$ on the space $\mathcal{C}_c^\infty(0,\infty)$, the space of smooth functions with compact support in $(0,\infty)$. I want to prove that the ...
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### How do I prove this topological space is T2 and compact?

Let $\tau$ be a topology: $\tau=\{A \subset \mathbb{R}^2 | S^1 \subset(\mathbb{R}^2 \setminus A ) \} \cup\{\mathbb{R}^2\}$ where $S^1=\{ (x,y) \in \mathbb{R}^2 | x^2+y^2=1 \}$ Prove $\tau$ is ...
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### Separability of $C^2_b(\Omega)$

Let $\Omega \subseteq \mathbb R$ be open and bounded. Denote by $C_b^2(\Omega)$ the space of functions $f:\Omega\rightarrow \mathbb R$ such that $f, f', f''\in C_b(\Omega)$. Endow this space with the ...
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### What is the geometric interpretation of a seperable space?

What is the geometric interpretation of a seperable space? I know the definition of a seperable space and I can give some examples.
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