Questions tagged [separable-spaces]

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

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Separable metric spaces and closed cover

When I read a proof of theorem, there is a statement "Since $X$ is a separable metric space, then we can find a closed cover $\{F_n^k\}_{k\in\mathbb{N}}$ of $X$ such that diam$\{F_n^k\}<\frac{...
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Potential mistake on theorem regarding separability on Topology by James Dugundji

I think there must be a mistake on the following statement (from the book Topology by James Dugundji. Chap, VIII sec, 7) 7.2 Theorem (1). The continuous image of a separable space is separable. (2). [...
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Vitali covering lemma and choice

Wikipedia proof of Vitali covering lemma uses Zorn's lemma. I was able to replace part of the argument which uses Zorn's lemma with another argument(I′ll provide the idea without the details in the ...
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The concentration of a measure $\mu$ on a single point in case $\mu$ satisfies this special property

Let $(E, d)$ be a metric space and $\mu$ a Borel probability measure on $E$. Assume $\mu$ has a property that if $(E_1, E_2)$ is a measurable partition of $E$, i.e., $E_1 \cap E_2 = \emptyset$ and $...
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The separability of $\mathbb{N}$ vs. the metrizability of $\ell ^1$.

Good afternoon, currently I'm trying to find an intuitive example for a space $E$, which is not separable, such that the space of finite measures $\mathcal{M}_f (E)$ on it is not metrizable. In ...
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The space of bounded operators $L(X \rightarrow X)$ where $X=\ell_1$ is NOT seperable

How can one prove that the space of bounded operators $L(X \rightarrow X)$ where $X=\ell_1$ is NOT seperable? I am trying to think what can contradict seperability, but I didn't have any progress.
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How to prove the space $L^2\big([a,b],w(t)\big)$ is complete and separable?

Suppose $w(t)$ is a positive and measurable function on $[a,b]$. If $x(t)$ is a measurable real function on $[a,b]$ satisfying $$ \left\| x \right\|^2 =\int\limits_{\left[ a,b \right]}{w\left( t \...
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Exercise 10, Section 30 of Munkres’ Topology

Show that if $X$ is a countable product of spaces having countable dense subsets, then $X$ has a countable dense subset. My attempt: Fix $p=(p_n)_{n\in \Bbb{N}} \in \prod_{n\in \Bbb{N}} X_n$. Let $...
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Every open subspace of a separable topological space is separable

I have encountered this well-known result when reading about separable spaces. Let $(E, \tau)$ be a separable topological space. Let $X \in \tau$ and $\tau_X$ its subspace topology. Then $(X, \tau_X)$...
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Metric space $(X,d)$ is separable if and only if $(X,m)$ is separable, where $ad(x,y)\le m(x,y) \le bd(x,y)$ for any $x,y \in X$ and $a,b>0$.

Given two metric spaces $(X,d)$ and $(X,m)$ be two metric spaces such that there exists $a,b>0$ such that $$ad(x,y) \le m(x,y) \le bd(x,y)$$ for any $x,y\in X$. Show that $(X,d)$ is separable if ...
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Is Probability Measure always tight?

We know that probability measures are tight if the metric space is separable and complete. Here tight means there exists a compact set in that metric space say $K$ such that $P(K) > 1- \epsilon$. I ...
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An easy question about the separability of $L^p(X,\mathcal{A}, \mu)$ space, where $p\in [1,+\infty)$

The theorem we have to prove is the following: Theorem 1. Let $(X,\mathcal{A},\mu)$ be a measure space such that $(i)$ the space $(X,\mathcal{A})$ is separable, that is exists a countable collection $\...
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Intersection of separable metric spaces endowed with the sum of metrics

We are given two metric spaces $(\mathfrak X_1,d_1)$ and $(\mathfrak X_2,d_2)$ which we assume separable and such that $\mathfrak X:= \mathfrak X_1 \cap \mathfrak X_2 \neq \emptyset$. Define $d(x,y):= ...
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Borel $\sigma$-algebra of separable $T_{3 \frac 1 2 }$ space generated by bounded continuous functions

A topological space $\Omega$ satisfies the $T_{3\frac 1 2}$ separation axiom if for every $A \subset \Omega$ closed, and every $x \in \Omega \setminus A$, there is a continuous function $f : \Omega \...
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Constructing sequence of nested closed sets in separable $T_{3\frac 1 2}$ space

Let $X$ be a separable $T_{3\frac 1 2}$ space. In other words, for every closed $A \subset X$ and every $x \in X \setminus A$, there is a continuous function $f : X \to [0,1]$ for which $f(x) = 0$ and ...
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Every normed space with countable dimension is separable.

I am trying to figure out if this reult from functional analysis is true. "Every normed space with countable Hamel dimension is separable." I know that this hold if the space is of finite ...
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The Span of a Countable Set is always Separable?

I wonder if the span of a countable set is always a separable subspace with the norm topology? It feels like a trivial question but how does one show this? Namely, I start of with a countable set $\{ ...
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Prove that $A(\Delta)$ is a separable space

Let $\Delta$ be the unit closed disk in the complex plane and let $A(\Delta)$ be the family of complex functions which are continuous in $\Delta$ and analytic in the interior of $\Delta$. Now, ...
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Reflexivity and Separability of $L^{\infty}$ and $L^{1}$

I am currently trying to understand the following proposition: Let $\mu$ be a Radon measure on $\mathbb{R}^n$. Then: (1) $L^{\infty}(\mu)$ is neither reflexive nor separable (2) $L^{1}(\mu)$ is not ...
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Apparent contradiction between weak metrizability results

I will fix $E \doteq \ell^2$, an infinite dimensional space, with separable dual. In here, there was a discussion that the weak topology of $E$ is not metrizable. By a result given in here, the disk $...
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Why is $\inf_{h \in E^\star}\{\|f-h\|_1+[h]^2\}= \min _{h \in E^\star} \{\|f-h\|_1^2+[h]^2\}$ in Brezis's solution of Ex 3.27.4?

I have a problem understanding the solution of Ex 3.27.4 in Brezis's book of Functional Analysis. Below is the Ex 3.27.4: Let $E$ be a separable Banach space with norm $\|\cdot\|$ . The dual norm on $...
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$L_{\infty}(U)$ is not separable, where $U\subset\mathbb{R}^N$ open

Let be $U \subset \mathbb{R}^N$ open. Prove that $L_{\infty}(U)$ is not separable. I tried to divide the open set $U$ with a uncountable collection of open balls origin centered in $\mathbb{R}^N$ like ...
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Can this set be not separable ? Can it be not compact?

Let $(\mathcal X,\Sigma_X)$ be a measure space, let $\mathcal S_X=\{ q : (\mathcal X, \Sigma_X, q)\text{ is a probability space}\}$, i.e. the simplex associated to $(\mathcal X,\Sigma_X)$. We equip $\...
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Construct an equivalent and strictly convex norm on a separable Banach space whose dual norm is also strictly convex

I'm doing Ex 3.27 in Brezis's book of Functional Analysis. Let $(E, | \cdot |)$ be a separable Banach space and $(E', \| \cdot \|)$ its dual space. Let $(a_n)$ be a countable dense subset of the ...
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Is the dual of a strictly convex Banach space strictly convex?

I'm doing Ex 3.27.4 in Brezis's book of Functional Analysis. The question leads to below statement for which I don't know if it's true or not. Let $(E, | \cdot |)$ be a strictly convex Banach space. ...
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Is this a typo in Brezis's Ex 3.24?

I'm doing Ex 3.24 in Brezis's book of Functional Analysis. The purpose of this exercise is to sketch part of the proof of Theorem 3.29, i.e., if $E$ is a Banach space such that $B_{E}$ is metrizable ...
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Prob. 5 (a), Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Every metrizabe separable space is second-countable

Here is Prob. 5 (a), Sec. 30, in the book Topology by James R. Munkres, 2nd edition: Show that every metrizable space with a countable dense subset has a countable basis. My Attempt: Let $X$ be a ...
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Brezis's Ex 3.22: If $E$ is reflexive, there is a sequence of norm $1$ that weakly converges to $0$

I'm doing Ex 3.22 in Brezis's book of Functional Analysis. Let $E$ be an infinite-dimensional Banach space satisfying one of the following assumptions: (a) $E'$ is separable (b) $E$ is reflexive. ...
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There is an isometry from a separable normed space $E$ into $\ell^\infty$

I'm doing Ex 3.20.2 in Brezis's book of Functional Analysis. There is an isometry from a separable normed space $E$ into $\ell^\infty$. My solution is different from the author's. Could you have a ...
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Inequivalence metrics generate same Borel $\sigma$-field.

I want to find an example of metric $d$ for $[0,1]$ so that $(d,[0,1])$ is a separable metric space; $d$ is not equivalent to usual metric, but the Borel $\sigma$-field is same as the usual one. Here ...
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Is there a nontrivial but short example where the separable degree is strictly less than the degree of a field extension?

I consider the following theorem: Let $E\supset F\supset k$ be a tower. Then $$[E:k]_s=[E:F]_s\cdot [F:k]_s.$$ Furthermore if $E$ is finite over $k$ then $[E:k]_s$ is finite and $$[E:k]_s\leq [E:k].$$ ...
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If $O_a=\{f \in L^\infty(\Omega):\|f-u_a\|_\infty<\frac{1}{2}\}$ then $O_a \cap O_b=\emptyset$ if $a \neq b$

This is part of the proof of Remark 9 ($L^\infty(\Omega)$ is not separable) in the French version of Brézis book. Let $\Omega$ be a open subset of $\mathbb{R}^n$. For each $a \in \Omega$ fix $r_a< \...
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$L^p(\Omega)$ is separable for $1 \leq p <\infty$

Following the Brézis book in French we have the following proof that $L^p(\Omega)$ is separable for $1 \leq p <\infty$. I have a doubt in the construction of the function $f_2$. Following the ...
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Any nonempty irreflexive Banach space contains a separable irreflexive Banach subspace - is my proof correct?

It is asserted in a proof I am following that: Let $X$ be a nonempty, irreflexive Banach space. By the Eberlein-Smulian theorem, $X$ contains a subspace $S$ that is Banach, irreflexive and separable. ...
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The normed space of convergent sequences of real numbers is separable (proof verification)

Let $c$ denote the normed space of convergent sequences of real numbers with norm $$||(x_1, x_2, x_3, \cdots)|| = \sup \{ |x_n| : n \in \mathbb{N} \}.$$ Note that $c$ is a Banach space. To show that $...
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The idea behind the proof that if $E'$ is separable, then so is $E$

I'm reading the proof of below Theorem 3.26 in Brezis's book of Functional Analysis. A similar proof can be found here. However, the idea behind the construction of such a countable dense subset is ...
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Baire space which doesn't contain a dense completely metrizable subspace

Let us work with separable metrizable spaces. It's known that if $X$ contains a dense completely metrizable subspace then $X$ is Baire. Moreover, if we assume that $X$ embeds into any space as a Borel ...
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Separable topological space where the dense subset remains dense after deleting finitely many points

Let $X$ be a separable topological space. By definition, it has a countable dense subset $S$. Suppose $S$ is infinite. My question is: What are the examples of such $X$ satisfying the property that ...
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Right-Continuous Martingale Separable-Valued

Let $(\Omega,\mathcal{F}, \mathbb{P})$ be a probability space and, for a Banach space $E$, let $L^0(\mathbb{P}; E)$ be the space of all $\mathbb{P}$-random variables, meaning $X\in L^0(\mathbb{P}; E)$ ...
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Is the following a valid countable dense subset of (0,1) the open unit interval?

Let $X = \mathbb{R} $, $D = \mathbb{Q}$ and $I = (0,1)$. We know that $X$ is a complete metric space and $D$ a countable dense subset, thus making $X$ separable. First of all, I do know that a ...
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Why does this inequality imply non-separability?

In [Da Prato, Giuseppe, and Jerzy Zabczyk, Stochastic equations in infinite dimensions. Cambridge University Press, 1992] p. 23 the authors give the following explanation of why the space $L(U,H)$ of ...
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Strongly zero-dimensinal spaces that are homeomorphic to other spaces under certain conditions

According to a theorem from van Engelen: Homogeneous zero-dimensional absolute Borel sets (1986): Theorem: If $X$ is a separable metrizable zero-dimensional absolute $F_{\sigma\delta}$ that is nowhere ...
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Proof $X$ separable $\Rightarrow (B^*, \tau_{pt})$ is a first-countable space: show subsets are open?

Lemma. Let $(X,\tau)$ be a topological space and $x\in X$. Let $\mathcal{B}\subseteq \tau$ be closed under finite intersections and let $x\in U, \forall U\in\mathcal{B}$. Then $\mathcal{B}$ is a local ...
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About separable (separability) [closed]

Here is the problem. what are the difference between separable on set $(\mathbb{R},\mathbb{Q},{\rm etc})$ and separable on space $((\mathbb{R},\tau),(X,\tau))$ with $\tau$ as topology? is there an ...
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Proof of $\ell_\infty$ is not separable

Let $\ell_\infty=\{\bar{x}=(x_1,x_2,x_3,\dots) \,:\,x_n\in \mathbb{R}, \sup_{n\in\mathbb{N}} |x_n|<\infty, n\in \mathbb{N} \}$ and $$d_\infty(\bar{x},\bar{y})=\sup_{n\in\mathbb{N}} |x_n-y_n|,$$ for ...
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1 answer
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Relating to a topology and its basis.

Exercise. Show that for the topology $\tau$ in $\mathbb{R}$ with the subbasis sets $A = \{y \in \mathbb{R} : x \leqslant y\}$ and $\mathbb{R} \backslash A = \{y \in \mathbb{R} : x \not\leqslant y\}$as ...
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$Z$ normed space : $\forall X\in SB\exists Y\subset Z : X,Y$ isomorphic. Show $ \exists c>0 \forall X \in SB \exists Y \subset Z : X ,Y$ c-isomorphic.

I am currently studying a functional analysis course as part III of the Cambridge Tripos. In the course we were quoted the result: Let $SB$ be the class of separable Banach spaces over $\mathbb{R}$. ...
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Separable Bnach space implies that the unit ball is separable [duplicate]

if someone could give me an idea of ​​how to carry out this demonstration I would greatly appreciate it, let E be a Banach space, demonstrate E is separables $\Rightarrow$ The closed unit ball $B_E=\{...
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Separable Banach space, unit ball, unit sphere demonstration [closed]

hello I have the following question, about a Banach space E, prove they are equivalent: 1.E is separables 2.The closed unit ball $B_{E}=\{x\in E :||x||\leq 1 \}$ is separables The closed unit ball $...
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sum of separable norms is separable

Let $X$ be a linear space and let $\|.\|_1$, $\|.\|_2$ be two norms on $X$ such that $X_i:=(X,\|.\|_i)$ is separable for $i=1,2$. Then also $Y:=(X,\|.\|_1+\|.\|_2)$ is separable. I tried the following:...
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