# Questions tagged [polish-spaces]

For questions involving Polish spaces, that is, separable and completely metrizable topological spaces.

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### Homeomorphism between $\mathbb N^\infty$ and a closed subset $\mathbb M$ of $(\mathbb N^\infty)^\infty$

Trying to figure out a proof of a lemma that I'm reading in Stochastic Relations by Ernst-Erich Doberkat. The Baire space, denoted $\mathbb{N}^\infty$, is the infinite product of the natural numbers. ...
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### Total variation and probability measures

I'm currently reading some notes about optimal transport and here there is the definition of total variation:$$\Vert\mu\Vert_{TV}=2\sup\vert\mu(A)\vert$$where $\mu$ is a probability measure of the ...
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### Suslin sets and projections

In a paper I am reading about optimal transport (we are in a proper metric space, https://arxiv.org/pdf/2004.08934.pdf, page 49 at the end), there's written in a proof "Being (this set) the ...
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### Projection of an Analytic Set is Analytic

Let $\mathscr{I}$ be the family of all finite sequences in $\mathbb{N}$ and let us use Greek letters $\alpha,\beta\ldots$ to denote elements of $\mathscr{I}$. Given a paved space $(X,\mathscr{X})$, a ...
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### $(X, \mathcal{T})$ Polish and $F \subseteq X$ closed $\implies$ topology generated by $\mathcal{T} \cup \{ F \}$ is Polish

I am trying to understand the proof of this statement (Kechris, Classical Descriptive Set Theory, p. 82). (13.2) Lemma. Let $(X, \mathcal{T})$ be Polish and $F \subseteq X$ closed. Let $\mathcal{T}_F$...
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### Second-countable, analytic, completely Baire spaces

First a few definitions. A topological space $(X,\tau)$ is said to be: a Polish space if it is separable and completely metrizable. Analytic, if there exists a surjective continuous map from a (Borel ...
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### Uncoutable Borel Set of Polish Space Contains Cantor Set: Proof Step

Currently, I am struggling to fully understand the following theorem from Alexander Kechris' book “Classical Descriptive Set Theory”: The last sentence is unclear to me. Why is the homeomorphic copy ...
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### On gaps in the dimension of subspaces

Let $P\subseteq\Bbb N\cup\{\infty\}$, with $0\in P$. Does there exist a "nice" space $X_P$ such that $X_P$ has a subspace of dimension $n$ iff $n\in P$? By nice I mean nice enough that there'...
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### Are Radon measures on Polish spaces $\sigma$-finite?

If $\Omega$ is a Polish space and $\mu$ is a Radon measure on $\Omega$ (i.e. an inner-regular Borel measure), is $\mu$ $\sigma$-finite? I know that Radon measures in general need not be $\sigma$-...
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### On positive dimensional Polish spaces in which every compact set has empty interior

A standard characterization of the Baire space is that is the only nonempty, zero dimensional, Polish space in which every compact set has empty interior (up to homeomorphism of course). I'm ...
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### Lusin’s Theorem for Polish spaces with infinite Radon measure

I’m working on the following exercise in Klenke’s Probability Theory: A Comprehensive Course (Exercise 13.1.3), which asks us to prove the following generalization of Lusin’s Theorem: Let $\Omega$ be ...
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### Polish group actions: if an orbit is non-meager in itself, it is a Baire space?

Assume that $G$ is a Polish group continuously acting on a Polish space $X$. Let $x \in X$ be a point such that $G \cdot x$, the orbit of $x$, is non-meager in its relative topology. I would like to ...
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### Countable comeager set in an (uncountable) non perfect polish space

Is it possible to have an uncountable and non perfect Polish space with a countable comeager set? Furthermore, is it possible for this space to have a comeager collection of isolated points? I've been ...
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If I have a point $x \in S$ where $S$ is some Polish Space with metric $d$, a closed set $F \subset S$, and $inf\;\{d(x,f)|f\in F\}=0$, then why can we say that there exists $f' \in F$ such that $d(x,... • 1,856 2 votes 0 answers 73 views ### Dense subsets in the hyperspace of compact sets Let$X$be compact and Polish (I'm thinking of$X=[0,1]$, but I guess maybe the same answer holds for every compact Polish space) and let$\mathbf{K}(X)$be the hyperspace of compact subsets of$X$, ... • 2,897 3 votes 1 answer 54 views ### Connectedness and complexity in Polish spaces I was wondering: How complex can connected subsets of Polish spaces be? Are there connected non-Borel subsets of a Polish space? Given$X$Polish space (not totally disconnected), does it have proper ... • 2,133 5 votes 1 answer 130 views ### Total disconnection and zero-dimension in Polish spaces First of all Polish spaces are completely-metrizable, separable topological space and by zero-dimensional Polish space I mean that the Polish space has a (countable) basis made of clopen sets. It is ... • 2,133 4 votes 1 answer 172 views ### Does proving that closed subset of Polish space is Polish require axiom of countable choice? Let$C$be a closed subset of polish space$P$. It is trivial that$C$is also completely metrizable, but how do we prove that$C$is separable? I came up with this method: We can prove that separable ... • 1,142 0 votes 0 answers 112 views ### Why is a Polish space a standard measurable space? A measurable space ($\Omega$,$\mathcal F$) is called a standard measurable space if it is Borel isomorphic to one of the following measurable spaces:$(\langle 1, n\rangle,\mathcal B(\langle 1, n\...
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Let $X$ be a separable metric space and let $Prob(X)$ be the set of Borel probability measures on $X$, equipped with the weak* topology. Here are the facts that I know: The collection \$\{\lambda\in ...