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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25 views

If $X$ is separable then $X^∗$ is separable in all topologies $\tau$ such that $(X^∗,\tau)^∗ =X$?

Let $(X,\|.\|_{X})$ be a separable Banach space and the associated dual space is denoted by $X^*$. By $w^*$ we shall indicate the weak$-*$ topology on $X^*$. Let $B_{X^∗}= \{x^∗ \in X^∗ : \|x^∗\|_{X^∗...
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Separable space of a topological [closed]

Let $ X $ be an uncountable set and fix $ a \in X $. Define $ T = \{ G \subseteq X : a \in G\} \bigcup \{\emptyset\} $. Is the topological space $(X, T)$ separable?
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34 views

Is this topological space separable? [closed]

Let $X$ be an uncountable set and $a \in X$. Let $\tau=\{ G \subseteq X :a \in G \} \cup \{\emptyset\} $ and $\tau'=\{ G \subseteq X :a \not \in G \} \cup \{X\} $. Questions: a) Is $(X,\tau)$ a ...
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Proof check: Subset of separable space is separable

I want to know if my proof is correct and, if so, if it could be simpler. I think the last two lines aren't fully convincing. Let $X$ be a separable metric space. Prove that every subset of $X$ is ...
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1answer
47 views

Show that if $X$ is compact metrizable then $C(X)$ is separable.

Let $X$ be a metrizable compact space. I want to show that $C(X, \mathbb{C})$ is separable in the uniform topology. Attempt: By our assumption, $X$ is separable, so we can pick a countable dense ...
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1answer
17 views

Separability of a space implies existence of an element with positive measure neighborhood.

Suppose $E$ is a separable Hilbert space and $X$ and $X'$ are measurable functions from $(\Omega,\mathcal{F},\mathbb{P})$ into $E$ such that $$\mathbb{P}\left( X-X'\neq 0 \right)>0.$$ I want to ...
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33 views

The space of bounded linear map from $L^2(\mathbb{R})$ to $L^2(\mathbb{R})$ is not separable.

I found an article arguing that the space of bounded linear maps $E=\mathcal{L}(U,U)$ is not separable where $U=L^2(\mathbb{R})$. The argument goes as follows. Consider the isometry bounded linear ...
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25 views

Seconed countable space is separable and first countable

Excuse me can you see this question Show that every seconed countable space is separable and first countable I tried on it but i am not sure , I get / Let(X,T) be seconed countable space so there ...
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Is my proof of the separability of the closure of the range of a compact operator correct?

I want to prove the following remark (found in https://www.uio.no/studier/emner/matnat/math/MAT4400/v19/pensumliste/ela-190523.pdf) My attempt: Since $T$ is a compact operator, $T(B)$ is precompact ...
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44 views

If $X$ is separable, $Y$ is $T_2$ and $f: X \to Y$ is continuous, then $f$ is decided by the values of a countable dense set

The Question is to prove that if $X$ is a separable space, $Y$ is a $T_2$ space, and $f: X \to Y$ is a continuous function, then $f$ is decided by the values of a countable dense set. I tried ...
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Prob. 16 (a), Sec. 30, in Munkres' TOPOLOGY, 2nd ed: The product space $\mathbb{R}^I$, where $I = [0, 1]$, is separable

Here is Prob. 16 (a), Sec. 30, in the book Topology by James R. Munkres, 2nd edition: Show that the product space $\mathbb{R}^I$, where $I = [0, 1]$, has a countable dense subset. My Attempt: ...
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Prob. 15, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: The subspace $\mathscr{C}(I,\mathbb{R})$ of $\mathbb{R}^I$ with uniform metric is separable

Here is Prob. 15, Sec. 30, in the book Topology by James R. Munkres, 2nd edition: Give $\mathbb{R}^I$ the uniform metric, where $I = [0, 1]$. Let $\mathscr{C}(I, \mathbb{R})$ be the subspace ...
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Prob. 11, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: A continuous image of a Lindelof (separable) space is Lindelof (separable)

Here is Prob. 11, Sec. 30, in the book Topology by James R. Munkres, 2nd edition: Let $f \colon X \rightarrow Y$ be continuous. Show that if $X$ is Lindelof, or if $X$ has a countable dense subset, ...
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206 views

Example of “almost metric” topological space

It is well known, that every subspace of separable metric space is separable. It is also known, this statement not to be true, if space is topological and not necessary metric. But I cannot find an ...
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28 views

Prob. 13, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Every collection of disjoint open sets in a separable space is countable [duplicate]

Here is Prob. 12, Sec. 30, in the book Topology by James R. Munkres, 2nd edition: Show that if $X$ has a countable dense subset, every collection of disjoint open sets in $X$ is countable. My ...
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28 views

Prob. 10, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: A countable product of separable spaces is also separable

Here is Prob. 10, Sec. 30, in the book Topology by James R. Munkres, 2nd edition: Show that if $X$ is a countable product of spaces having countable dense subsets, then $X$ has a countable dense ...
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51 views

Show that if the norms ∥⋅∥1 and ∥⋅∥2 are equivalent, then if space (X,∥⋅∥1) is complete, then the space (X,∥⋅∥2) is not necessary complete.

X is complete if Cauchy sequence converges to it's limit in X ∀ε>0 ∃N: ∀n>N∶ ∥xn-x∥1<ε So let (X,∥⋅∥1) – separable and xn - Cauchy sequence in (X,∥⋅∥1) and xn→x ∀ε>0 ∃N: ∀n>N∶ ∥xn-x∥1<ε We have ...
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Prob. 5 (a), Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Every separable metric space is second-countable

Here is Prob. 5, Sec. 30, in the book Topology by James R. Munkres, 2nd edition: (a) Show that every metrizable space with a countable dense subset has a countable basis. (b) Show that every ...
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Prove the compactness and second countability of a subset of $\Bbb{R}^{L^1(G)\times D}$

I'm trying to fulfill the details in a proof whose aim is to guarantee the existence of a compact space with certain properties, but in this case I had no success. Let $X$ be a metric space (actually,...
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1answer
17 views

Separating Distributions with A Plane

Is the following statement true? If so does there exist a proof or something similar? "Given two independent probability distributions, $P_{A}(X)$ and $P_{B}(X)$, neither of which is uniform, there ...
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47 views

Radially open set topology is separable?

Consider the topology of radially open sets on $\mathbb{R^2}$ : a set $S$ is radially open iff, for every $x\in S$ and every line $L\subset\mathbb{R^2}$ that contains $x$, $S$ contains an open segment ...
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27 views

Confusion about separability of the space $\ell^2$

I found the following result: $\ell^2$ is not separable. The proof use this claim as corollary. Let $(E, d)$ a metric space. If there exist $B\subseteq E$ uncountable such that for all $...
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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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61 views

prove that $D=\overline{\text{span}\{x_n:n \in \mathbb{N}\}}$ is separable

Let $E$ be a normed space. Let $\{x_n: n \in \mathbb{N}\} \subset E$. I need to prove that $D=\overline{\text{span}\{x_n:n \in \mathbb{N}\}}$ is separable. I've defined $A_n= \left\{ \displaystyle\...
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18 views

Topological Vector Space is separable if its dual space $X^*$ is separable?

Let $(X,\tau)$ be a Topological Vector Space such that the associated dual space $X^*$ is separable. Can we say that $X$ is separable ? I know that this property is valid for Banach spaces but for ...
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Relation between total boundedness and separabilty.

Definitions: A set $A$ in a metric space $X$ is said to be an $\epsilon$-net in $Y$ if for each $y\in Y,\exists a\in A$ such that $d(a,y)<\epsilon$.A set $A$ is said to be an universal net of $Y$ ...
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Proving a well known result of metric spaces without using Zorn's Lemma.

I saw a proof of the result using Zorn's Lemma: Statement: If $X$ is a metric space in which every uncountable set has a limit point,then $X$ is separable. The proof was a bit constructive and does ...
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1answer
58 views

Riesz representation for a countable family of functions

In reading this I have found the following result that I don't know how to prove precisely: Theorem: Let $(X,d)$ be a compact metric space and let $C(X)$ be the Banach space of real valued ...
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1answer
27 views

$Y\subset X$ is Lindelof iff Every $X$-open cover of $Y$ has a countable $X$-open subcover. [duplicate]

Suppose $(X,\tau)$ is a topological space and $A\subset X$,suppose $\{G_\alpha\}$ is an open cover of $A$ i.e. $A\subset \cup G_\alpha$,does there exist a countable subcover $\{G_{\alpha_n}\}$ of $A$....
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42 views

Give an example of a separable topological space where uncountable set does not necessarily have limit point.

In metric space there is a well-known result that $X$ is a separable metric space $\implies$Every uncountable set in $X$ has a limit point.My question is does the result hold if $(X,\tau)$ is ...
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109 views

$L^p$ space is separable if and only if measure space separable

I am currently dealing with the following question: Let $(E,\mathcal{A},\mu)$ be measure space, $p\in[0,+\infty)$. Show that $L^{p}(\mu)$ has a countable dense set iff there exists $(A_n)\in \mathcal{...
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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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48 views

Proving $l^{\infty}$ not separable.

I have proved the fact that $l^\infty$is not separable.I just want to verify whether my proof is correct.Suppose $A$ is a countable set in $l^\infty$.We have to show $A$ is not dense.Suppose $A=\{r_1,...
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Reflexive space and separable spaces

I read the definition of reflexive space and separable spaces, but I did not understand what was the point of studying them. What is a reflexive Banach space? What is its interest in functional ...
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53 views

Separability of $\mathbb R^{[0,1]}$ in the product topology

Munkres has the following exercise: Show that the product space $R^I$, where $I=[0,1]$, has a countable dense subset. If $J$ has cardinality greater than $2^\mathbb{N}$, then the product space $\...
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31 views

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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$S=\operatorname{cl}_S(A), T = \operatorname{cl}_T(B) \implies \operatorname{cl}_{S\cup T}(A \cup B) = S \cup T$

Let $X$ be a topological space with subsets $A \subseteq S \subseteq X$ and $B \subseteq T \subseteq X$ such that $S=\operatorname{cl}_S(A), T = \operatorname{cl}_T(B)$. Prove that $\operatorname{cl}...
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1answer
41 views

Show that $Y $ is separable.

Let $(X,\|.\|) $ be a banach space and $(A_n)_n $ be a famile of separable subset of $X$. Let $Y $ be a linear subset of $X $ which is generated by the union of $(A_n)_n$. Show that $Y $ is ...
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45 views

Examples of Dense\Not Dense sets

Let the set of eventually zero sequences $c_{00} = \{x = (x_1, x_2, . . .) : x_n = 0 \ \text{for all but finitely many} \ n \}$ where $x_i$ are real numbers. (a) Prove that $c_{00}$ is dense in $l^p, ...
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1answer
37 views

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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45 views

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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51 views

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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57 views

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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55 views

Questions about giving an example in subspace topology exercise

I just want to know if my guess for the following question is correct" Give an example of a separable Hausdorff space $(X,T)$ that has a subspace $(A, T_{A})$ is not separable. I am guessing $X=\...
3
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1answer
64 views

Differentiating term by term in a Banach space: how to justify it?

After looking at this question, I am now wondering if the theorem proven in the first answer below can be generalized to a Banach space. See here for my attempt. But before doing that, I have the ...
3
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1answer
69 views

Separability of bounded operators on normed spaces

In the case of Hilbert spaces $\mathcal H$, $\mathcal G$ it is known that the Banach space of all bounded operators $\mathcal B(\mathcal H,\mathcal G)$ is norm-separable if and only if $\operatorname{...
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55 views

A few questions concerning shearlets

I am currently reading Compactly supported shearlets are optimally sparse by Kutyniok and Lim and have a few small questions, which I have to give a talk about. At our university, this seminar is ...
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1answer
54 views

What are the application of result “Every compact metric space is separable”? [closed]

What are the application of result "Every compact metric space is separable"? I wanted to do some exrecise problem which uses the result Every compact metric space is separable so that I can ...

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