Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [separable-spaces]

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

1
vote
2answers
36 views

Schauder basis $\implies $ Separable for non translation invariant metric linear spaces

It is fairly straightforward to prove that over a normed space $ (V,\| \cdot \|)$ the existence of a Schauder basis $ \{ e_n\}_{n=1}^\infty$ implies the separability of the space. I was however ...
1
vote
1answer
35 views

Prove that if a metric space (X,d) is separable, then its completion ($\hat{X}, \hat{d}$) is separable.

Prove that if a metric space $(X,d)$ is separable, then its completion $(\hat{X}, \hat{d})$ is separable. So we want to show there exists a countable dense subset in $\hat{X}$. My attempt: Suppose a ...
1
vote
1answer
25 views

Separability of the Space of Random Banach Space Valued Functions

Suppose I have a separable Banach space $X$. In many texts on infinite dimensional analysis, one considers the set $L^2(\mathbb{P};X)$, $X$ valued random variables that have finite second moment, $\...
5
votes
2answers
68 views

Separability and the Nagata-Smirnov Metrisation Theorem

Definitions: Let $X$ denote a topological space throughout. If all singleton subsets of $X$ are closed, then we call $X$ Fréchet. If, given any closed subset $C \subset X$ and any point $x \in X - C$...
0
votes
2answers
25 views

Example of a space that is separable but not complete

I know that in general metric space $X$ can be separable without being complete. What's a good example?
2
votes
2answers
128 views

Find bijection $\ell^{\infty} \to [0,1]$

In a comment on the first answer to this question, @Nate Eldredge stated that "For instance, there is a metric on $\ell^{\infty}$ that makes it isometric to $[0,1]$. How would that look like? ...
0
votes
1answer
18 views

Question on completing proof separability in metric space implies countability [duplicate]

Let $X$ be a metric space. I want to show that: $X$ separable $\Rightarrow X$ second countable. What I have been able to show thus far: Let $Y$ be the countable dense subset in $X$. I have shown ...
0
votes
1answer
49 views

Proof that l2 has a countable and dense subset

this is my first question and I hope I don't make any relevant mistakes. For a little bit of context, in my real analysis homework I have the following problem. Show that the subset D $\subset$ $l_2$...
1
vote
1answer
24 views

Orthogonal Projection in Hilbert Space to prove weak convergence

Let $H$ be a hilbert space and $(h_{n})_{n\in\mathbb{N}}$ be a bounded sequence in $H$. Define $H_{0}:= \text{cl}(\text{span}(h_{1},h_{2},...))$. Then, $H_{0}$ is a separable space since the set of ...
0
votes
0answers
36 views

Topology on $\mathbb R$ with a disconnected subset $A$ where the intersection of seperations of $A$ with $\mathbb{R}-A$ is not $\varnothing$?

$\newcommand{\R}{\mathbb{R}}$ Hello, I was wondering how to find a topology on $\R$ and a disconnected subset $A$ such that every pair of sets, $U$ and $V$, that is a separation of $A$ in $\R$ ...
0
votes
1answer
18 views

to show cardinality in separable Hilbert space setting

enter image description here the above exercise in Conway's Functional Analysis book to me, the setting is too rough so i have no idea to step forward could you help me to start this proof? or just ...
0
votes
1answer
18 views

Why is the subspace of uniformly rho continuous functions separable?

I am reading through the book of Kosorok on Empirical Processes and I got stuck on a statement that seems to be clear to the author. To me, it is not clear at all. In chapter 6.1 page 87 he states ...
1
vote
0answers
5 views

Required Number of LU Factorization is $1$ If the System is Separable

I am reading the paper about the deformation transfer, which is to implant the vertex transformation at one mesh into another one. At page 4 of the paper, it has stated as: "... Furthermore, the ...
0
votes
0answers
16 views

What are useful application of that “non-commutativity of quaternion multiplication prevents the transition of changing ij = +k to ji = −k”?

I read here I don't understand well this conclusion The non-commutativity of quaternion multiplication prevents the transition of changing ij = +k to ji = −k. I understand that this is not a real ...
0
votes
0answers
21 views

Decomposition theorem for topological dimension

I am trying to prove the decomposition theorem for the topological dimension: It is a known result (e.g. Engelking) that for a non-empty separable metric space $X$, the small inductive dimension $...
0
votes
2answers
27 views

Is there a metric for which the open unit interval is complete?

Let, $I= (0,1)$ It is well known that $I$ is not a complete with respect to the Euclidean metric $(x,y)\mapsto |x-y|$. However, $(I,|\cdot|)$ is separable. Question: Can we find a metric $d: I\...
1
vote
1answer
18 views

Completely disconnecting a separable metric space by removing a sequence of countable dense subsets

I am wondering if given a separable metric space $X$, it is possible to totally disconnect $X$ by repeatedly removing countable dense subsets. For example, let $I_1$ be a countable dense subset of $X$...
2
votes
1answer
91 views

How to show that the space of probability measures on $\mathbb{R}$ is separable under Lévy metric

The Lévy metric between distribution functions $F$ and $G$ is given by: $$\rho(F,G) = \inf\left\{\epsilon : F(x-\epsilon)-\epsilon\leq G(x)\leq F(x+\epsilon)+\epsilon\right\}.$$ Another way to ...
3
votes
1answer
62 views

The spaces $\ell^p, \; 1 \leq p < + \infty$ are separable. On the other side, $\ell^\infty$ is not.

Exercise : Show that the spaces $\ell^p, \; 1 \leq p < + \infty$ are separable. Attempt : In order to show that $\ell^p$ is separable for $\ell^p, \; 1 \leq p < + \infty$, we need to work ...
4
votes
1answer
38 views

Is the family of all continuous functions from a compact metric space to a separable complete metric space separable?

Let $(X,d)$ be a separable complete metric space. Let $K$ be a compact metric space. I denote by $C(K,X)$ the set of continuous functions $f:K\rightarrow X$ with the metric $\rho:C(K,X)\times C(K,X)\...
0
votes
1answer
40 views

$X$ is connected and separable. $X=Y\times Y$. Does $Y$ has to be also connected and separable?

$I$ is a finite set. It is not hard to see that, if $X=\prod_{i\in I}Y_i$ is separable, then $Y_i$ does not have to be separable. But for this special case such that $Y_i=Y_j \ \forall i,j\in I$, I ...
4
votes
1answer
22 views

Confusion about real separable normed space problem

Suppose that $X$ is a real separable vector space, and $W$ a closed linear subspace of $X$. Show that there exists a sequence $(z_j )\in X$ such that $z_{j+1} \notin W_j := \mathrm{span} \ W \cup \...
4
votes
1answer
53 views

Constructive proof of Banach-Alaouglu

Is there a constructive (i.e. not using Axiom of choice, and at most Axiom of dependent choice) proof of Banach-Alaoglu theorem in the case of separable Banach spaces. Even if it is needed assume that ...
0
votes
2answers
19 views

Explain the statemets related to separable spaces.

Separable spaces are "simpler" than nonseparable ones-KREYSZIG Separable space $X$ is not "too big" in the sense that we can approach each element of space $X$ through a sequence of elements of a ...
0
votes
2answers
32 views

Altering separable space's definition

A separable space has by definition a countable sub-space that is dense. What if we replace "dense" by "its completion is dense" What about "its completion is the whole space"? Do such spaces have ...
0
votes
1answer
27 views

A boolean algebra is countable iff the Stone space is separable

Let $B$ a Boolean Algebra and $St(B)$ its stone space. I want to prove that $B$ is countable if and only if $St(B)$ is separable (i.e. there is a countable dense subset). For the first implication: ...
0
votes
1answer
28 views

Continuous injections of nice spaces into $\ell^2$

If X is a separable, first countable Hausdorff space, then does there exist a continuous map from X to $\ell^2$? Intuitively put, can you always fill in the holes of nice spaces?
2
votes
3answers
115 views

Proof that the following map $\Phi:\ell^1\to(\ell^\infty)'$ is not surjective

I am working on the dual spaces of sequence spaces, and I want to show that the map $$ \Phi:\ell^1\to(\ell^\infty)',\qquad(\Phi y)(x)=\sum_{i\in\mathbb{N}}y_ix_i $$ is not surjective. I have already ...
0
votes
1answer
38 views

Is every separable locally compact metrizable topology induced by a Heine-Borel metric?

This is a follow-up to my question here. A metric has the Heine-Borel property if a set is closed and bounded with respect to the metric if and only if it is compact. Now if a metrizable topology is ...
1
vote
2answers
41 views

Showing that $Y = \cup_{n=1}^\infty Y_n$ is dense over the Hilbert $H$.

Source of question : Trying to prove that if the Hilbert $H$ has an orthonormal basis, then it is separable. Elaboration : Let $H$ be a Hilbert space and $\{e_n : n \in \mathbb N\}$ an orthonormal ...
2
votes
1answer
48 views

Proving that if the dual space $X^*$ of a Banach space $X$ is separable , then $X$ is separable

I have reading an answer post before , and there's something I do not understand . For each $f_n$ in the unit ball of $X^*$ , why there exist $x_n \in X$ with $\|x_n \| \le1$ such that $f_n(x_n) \ge \...
1
vote
1answer
36 views

Banach space with subset whose elements are at least $d\gt 0$ far from each other is not separable

Let $X$ be a Banach space, and $A\subseteq X$ subgroup, where $A$ is not countable, and there is some $d \gt 0$ such that for all $x,y \in A$: $||x-y||>d$. Prove that $X$ is not separable. My ...
0
votes
1answer
40 views

Similarity and Difference between Separable Space and Separated space?

Does separability and/or second countability implies $T_2$ or higher axiom sets? My intuition is "no". Even $T_0$ space can be separability and/or second countability?
1
vote
1answer
24 views

Properties of basis of regular open sets in separable spaces

I'm trying to prove that in any regular Hausdorff separable space, given a dense countable set D, for any open set $U$ such that $U=\text{int Cl}(U)$ it happens that $U=\text{int Cl}(U\cap D)$. I've ...
0
votes
1answer
80 views

Lindelöf and separable metric space [duplicate]

Let $(X,d)$ be a metric space. How to prove that every lindelöf metric space is separable?
0
votes
1answer
23 views

Continuity of the derivative on a separated interval

Suppose $f:[a,b]\rightarrow \mathbb{R}$ is a continuous function that is differentiable on the open interval $(a,c) \cup (c,b)$ for some $c\in [a,b]$. Show that if $\lim_{x\to c} f'(x)$ exists, then $...
1
vote
0answers
23 views

Separable $\Rightarrow$ Lindelöf for metric spaces without using second-countability [duplicate]

It is well-known that for metric spaces, being separable, strongly Lindelöf and second-countable are equivalent. I know how to prove the equivalence between separable and second-countable, and I guess ...
0
votes
0answers
29 views

Proving $L^p(R,L, λ)$ is separable by using step functions with finite support

I am trying to prove that $L^p(R,L, λ)$ with the $L^p$-norm is a separable space for $1 ≤ p < ∞$ by the following method. Now I know that if $f ∈ L^p$, then for any $ε > 0$, there is a step ...
3
votes
1answer
67 views

Show that $C(K,E)$ is separable

This is an exercise of the book Analysis III of Amann and Escher: Let $E$ a separable Banach space over $\Bbb K$ (being $\Bbb K=\Bbb R$ or $\Bbb K=\Bbb C$) and $K$ a compact metric space. Show that ...
1
vote
1answer
32 views

$L^∞(R,L, λ)$ Inseparable?

I know that $L^p(R,L, λ)$ with the $L^p$-norm is a separable space for $1 ≤ p < ∞$ Here if a normed space $X$ has a countable dense subset, then X is said to be separable. For example, R is ...
1
vote
3answers
68 views

Inseparable complete metric space with full Radon probability measure

I am looking for a simple construction of an inseparable complete metric space that carries a Radon probability measure with full support. A similar question was asked before although not in the ...
1
vote
1answer
52 views

please help me to visualize the open sets in $\tau$.

In "Foundation of Topology" by C.W Patty, given $$\begin{align}A :=& \ \{(x,y)\in \mathbb R^2:y=0\},\\ X:=& \ A\cup\{(0,1)\},\end{align}$$ the topology on $X$ is defined as $$\tau:= \ \{U\...
1
vote
1answer
83 views

Proof doubt: $\ell^\infty$ is not separable

I have stumbled upon a particular proof of a fact that $\ell^\infty$ is not separable which I can't quite grasp: For $M\subset \mathbb N$ define $\chi_M(n)=1$ if $n\in M$ and $0$ otherwise. $\...
1
vote
0answers
34 views

Linear separability in $\mathbb{R}^{1}$

Following Wikipedias definition, two disjunct sets $X_0, X_1 \subset \mathbb{R}^{n}$ are linearly separable, if there exists a $k \in \mathbb{R}$ and a vector $w \in \mathbb{R}^{n}$ such that the ...
4
votes
1answer
62 views

Is the Banach space of continuous functions $I \to \ell^2$ separable?

Inspired by this question. Consider the vector space $V$ of all continuous functions $I \to \ell^2$ for $I=[0,1]$ the closed unit interval and $\ell^2$ the Hilbert space of all square-summable ...
3
votes
2answers
72 views

Given $k$ points in $n$-dimensional space, is there always a continuous $n-1$ surface that can divide the points into two arbitrary groups?

Say we have $k$ points in set $P\mid x_i\in\mathbb{Z}^n$, such that $k=\mathcal{O}(n!)$. We now arbitrarily divide the points into two sets, $A$, $B$. Note that $A\cup B = P$ and $A \cap B = \...
0
votes
1answer
35 views

A countable open base implies that any open covering has a countable subcovering.

Proposition: If metric space $M$ has a countable open base, then any open covering of $M$ admits a countable subcovering. Definition: A collection of open subsets $\{U_i\}$ of $M$ is called an open ...
3
votes
1answer
65 views

Bounded sequence has Cauchy subsequence w.r.t. $ (x|y)_0:=\sum^\infty_{n=1} 2^{-n}\phi_n(x)\phi_n(y).$

Let $X$ be a separable reflexive real Banach space and let $(\phi_n)$ be a dense sequence in $$\{ \phi\in X'\,|\,\|\phi\|\leq 1 \}.$$ Consider in $X$ the scalar product $( \cdot| \cdot)_0 $ defined by ...
6
votes
2answers
112 views

If $X \times Y$ is Separable, are $X, Y$ Separable?

I would suspect the question in the title is false, but I could not think of a counterexample. The reason I am interested in this question concerns the various definitions of 'generalized manifolds.' ...
2
votes
1answer
61 views

Countably Compact, Separable, $T_1$, Connected Space that is not Compact

In Wilder's Topology of Manifolds, the following is stated on p. 43: "It is well known that not every countably compact, separable, connected space is compact." Hmm . . . I'm not sure just how well-...