# Questions tagged [separable-spaces]

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

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### 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 ...
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### 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 ...
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### $X$ 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### Lindelöf and separable metric space [duplicate]

Let $(X,d)$ be a metric space. How to prove that every lindelöf metric space is separable?
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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 $... 0answers 23 views ### Separable$\Rightarrow$Lindelöf for metric spaces without using second-countability [duplicate] It is well-known that for metric spaces, being separable, strongly Lindelöf and second-countable are equivalent. I know how to prove the equivalence between separable and second-countable, and I guess ... 0answers 29 views ### Proving$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 ... 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 ... 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 ... 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 ... 1answer 52 views ### please help me to visualize the open sets in$\tau. In "Foundation of Topology" by C.W Patty, given \begin{align}A :=& \ \{(x,y)\in \mathbb R^2:y=0\},\\ X:=& \ A\cup\{(0,1)\},\end{align} the topology onX$is defined as $$\tau:= \ \{U\... 1answer 83 views ### Proof doubt: \ell^\infty is not separable I have stumbled upon a particular proof of a fact that \ell^\infty is not separable which I can't quite grasp: For M\subset \mathbb N define \chi_M(n)=1 if n\in M and 0 otherwise. \... 0answers 34 views ### Linear separability in \mathbb{R}^{1} Following Wikipedias definition, two disjunct sets X_0, X_1 \subset \mathbb{R}^{n} are linearly separable, if there exists a k \in \mathbb{R} and a vector w \in \mathbb{R}^{n} such that the ... 1answer 62 views ### Is the Banach space of continuous functions I \to \ell^2 separable? Inspired by this question. Consider the vector space V of all continuous functions I \to \ell^2 for I=[0,1] the closed unit interval and \ell^2 the Hilbert space of all square-summable ... 2answers 72 views ### Given k points in n-dimensional space, is there always a continuous n-1 surface that can divide the points into two arbitrary groups? Say we have k points in set P\mid x_i\in\mathbb{Z}^n, such that k=\mathcal{O}(n!). We now arbitrarily divide the points into two sets, A, B. Note that A\cup B = P and A \cap B = \... 1answer 35 views ### A countable open base implies that any open covering has a countable subcovering. Proposition: If metric space M has a countable open base, then any open covering of M admits a countable subcovering. Definition: A collection of open subsets \{U_i\} of M is called an open ... 1answer 65 views ### Bounded sequence has Cauchy subsequence w.r.t. (x|y)_0:=\sum^\infty_{n=1} 2^{-n}\phi_n(x)\phi_n(y). Let X be a separable reflexive real Banach space and let (\phi_n) be a dense sequence in$$\{ \phi\in X'\,|\,\|\phi\|\leq 1 \}.$$Consider in$X$the scalar product$( \cdot| \cdot)_0 $defined by ... 2answers 112 views ### If$X \times Y$is Separable, are$X, Y$Separable? I would suspect the question in the title is false, but I could not think of a counterexample. The reason I am interested in this question concerns the various definitions of 'generalized manifolds.' ... 1answer 61 views ### Countably Compact, Separable,$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-...