Questions tagged [polish-spaces]
For questions involving Polish spaces, that is, separable and completely metrizable topological spaces.
111
questions
0
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1
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41
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The countability of the Cantor-Bendixson rank in Polish spaces
I've been studying the Cantor-Bendixson theorem and have some questions about the proof of the countability of the Cantor-Bendixson rank in Polish spaces. I would greatly appreciate your insights on ...
- 85
3
votes
1
answer
126
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Is this set countable for any function?
Working with CH I order ($\prec$) $^\omega\omega$ of order type $\omega_1$. I let $$f_\alpha:=\min_\prec\{f\in\hspace{1mm}^\omega\omega:\neg (f(n) \leq f_\beta(n)),\forall n\in\omega,\beta\in\alpha\}$$...
- 315
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213
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Can you construct a sequence to witness that this set isn't compact? [closed]
I know that if $K$ is a closed subset of $^\omega\omega=:\mathcal{N}$ and there exists an $f\in\mathcal{N}$ such that $K\subseteq \{g\in \mathcal{N}:g\leq f \}$ (where $g\leq f$ is pointwise), then $K$...
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2
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Are these sets of functions finite?
Can there be a set of functions $S\subseteq\{f \mid f:\mathbb{N}\to\mathbb{N}\}$ of cardinality $\mathfrak{c}$ (real numbers) such that the set $S_f:=\{g \in S \mid g(n)\leq f(n), \forall n \in \...
- 315
5
votes
1
answer
119
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When is this closed set compact
Apparently in the polish space $^\omega\omega$ a closed $K\subset\hspace{1mm}^\omega\omega$ is bounded and therefore compact if it is completely below some $f\in \hspace{1mm}^\omega\omega$ as in $K= \{...
- 113
4
votes
0
answers
77
views
Suslin measurable sets and the smallest field containing all analytic sets
Let $X$ be a Polish space. Recall that the Suslin operation is the operation $\mathcal{A}$ such that for any Suslin scheme $\{A_s : s \in \omega^{<\omega}\}$ of subsets of $X$, we have:
$$
\mathcal{...
- 8,387
3
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Extension of Borel map from a separable metric space to a Polish space
Suppose that $f:X\to Y$ is a Borel map from separable metric space $X$ to a $T_3$ space $Y$.
Does there always exist a Polish space $\tilde X \supseteq X$ and $T_3$ space $\tilde Y\supseteq Y$ and an ...
- 10.5k
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49
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Axiom of Choice and Borel determinacy for Polish space
Given a set $A$, Borel determinacy for $A$ is the theorem (of $\mathsf{ZFC}$) asserting that every Borel subset of $A^\omega$ is determined. That is, if I and II take turns playing members of $A$, and ...
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47
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Locally compact Polish space admits a proper metric [duplicate]
If $X$ is locally compact Hausdorff, then the following are all equivalent:
$X$ is second countable,
$X$ is metrizable and $\sigma$-compact,
$X$ is metrizable and separable,
$X$ is Polish.
I want to ...
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3
votes
1
answer
84
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Why is the set of probability measures not weak*-compact?
Let $M(X)$ be the set of probability measures on a Polish space $X$ with Borel $\sigma$-field. Further consider the properties of $M(X)$ when considered as members of the dual space of $Y:=C_b(X)$ - ...
- 839
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31
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Functional E convex and lower semicontinuous implies weakly lower semicontinuous in Wasserstein space
I have a certain functional $E : W_2 \rightarrow \mathbb{R}$, where $W_2$ is the 2-Wasserstein space (metric and separable).
Such functional is convex.
Now, can I state that if $E$ is strongly lower ...
- 21
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votes
0
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37
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Sufficient Conditions on Metric Space for Wasserstein Distance?
For a metric space $M$ and the Wasserstein 1-distance $W_1$, what qualifying assumptions do we need for $M$? I have seen we only need $M$ be compact, $M$ be a Polish space, or $M$ be a Radon space. ...
- 71
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35
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Comparing sigma-algebras generated by a random variable and its induced posterior
Consider two random variables $\tilde x$ and $\tilde y$ taking values in Polish spaces $X$ and $Y$ respectively. Let the (prior) distribution of $\tilde x$ be $\nu$ and the distribution of $\tilde y$ ...
- 1
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236
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Total variation distance: a relationship between a Polish space $(\mathcal{X}, d)$ and a measurable space $\left(\mathcal{X},\mathcal{A}\right)$
Introduction (part 1). In the following excerpts of Villani (2008) Optimal transport, old and new, Villani (i) defines the Wasserstein distance among two probability measures $\mu$ and $\nu$ on a ${\...
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Refine product topology to make Borel sets be clopen
I'm working on Exercise 2.28 in Prof. David Marker's notes http://homepages.math.uic.edu/~marker/math512/dst.pdf on refining the topology to make Borel sets clopen.
Question: Suppose $X$ is a Polish ...
- 346