Questions tagged [separable-spaces]
For questions about separable spaces, i.e., topological spaces containing a countable dense set.
536 questions
1 vote
0 answers
33 views
Prob. 7, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Is $\overline{S_\Omega}$ first-countable? second-countable? separable? Lindelof?
Here is Prob. 7, Sec. 30, in the book Topology by James R. Munkres, 2nd edition: Which of our four countability axioms does $S_\Omega$ satisfy? What about $\overline{S_\Omega}$? Our four ...
- 27.8k
1 vote
0 answers
146 views
Prob. 7, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Is $S_{\Omega}$ first-countable? second-countable? separable? Lindelof? [closed]
Here is Prob. 7, Sec. 30, in the book Topology by James R. Munkres, 2nd edition: Which of our four countability axioms does $S_{\Omega}$ satisfy? ... Here the four countability axioms are (i) first-...
- 27.8k
5 votes
1 answer
117 views
When is a metrisable Souslin space a Polish space?
Recall that a Polish space is a separable topological space which can be metrised by a complete metric, and that a Souslin space is any Hausdorff topological space $S$ admitting a continuous ...
- 48.1k
2 votes
2 answers
116 views
Metric space generated by non empty closed sets of $\mathbb{R}^n$ is separable
I was working on the set $\mathcal{X}=\left\{ F\subseteq\mathbb{R}^{n}|F \text{ is closed and nonempty on } (\mathbb{R}^n,\left\| \cdot \right\|) \right\}$ with the metric given by: $$\hat{d}(C,D)=\...
- 178
0 votes
1 answer
145 views
Is an infinite, separable Hilbert space tensor itself isomorphic to itself?
Suppose I have an infinite, separable Hilbert space $\mathcal{H}$ over $\mathbb{C}$ and bounded linear operators $D$ and $T$ on $\mathcal{H}$ such that $D$ is a diagonal operator given by $$\begin{...
2 votes
0 answers
55 views
Is atomic Hardy space $H_1(\mathbb{R})$ separable?
The atomic Hardy space $H_1(\mathbb{R})$ is space of functions in $L_1(\mathbb{R})$ of the from $$ f(x) = \sum_{i=0}^\infty \lambda_i a_i $$ where $(\lambda_i)_i$ is in $\ell_1$, and each $a_i$ is an ...
- 91
2 votes
1 answer
211 views
a connected locally separable metric space is separable
Is there a modern reference for a proof of the following? A connected locally separable metric space is separable. It is said to be from Paul Alexandroff's "Über die Metrization der im kleinen ...
- 2,900
1 vote
0 answers
38 views
Finite-rank operator convergence over separable Hilbert space
Suppose I have an operator $A$ on some separable Hilbert space $\mathcal{H}$, which might have infinite rank (the rank is defined as the dimension of the range). In other words, we take an orthonormal ...
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