# Tagged Questions

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### Measure on hyperspace of compact subsets

For a Polish space $X$, let $K(X)$ be the set of compact subsets of $X$. Given the topology with basis $\{K\in K(X):K\subset U_0, K\cap U_1\neq\emptyset,\ldots,K\cap U_n\neq\emptyset\}$ for open sets ...
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### Quotients of standard Borel spaces

Let $X$ and $Y$ be standard Borel spaces: topological spaces homeomorphic to Borel subsets of complete metric spaces. Given a surjective Borel map $f:X\to Y$, we get an equivalence relation ...
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### Is $\ell^\infty$ Polishable?

Consider $\ell^\infty$ as a subspace of the Polish space $\mathbb{R}^\omega$. It is easy to check that $\ell^\infty$ is not Polish in the subspace topology, as it is countable union of the compact ...
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### Strong measure zero sets and selection principles

A set of reals $X$ is strong measure zero if for any sequence of positive real numbers $( \epsilon_n ) _{n \in \omega }$ there is a sequence of open intervals $( a_n ) _{n \in \omega }$ which ...
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### sets without perfect subset in a non-separable completely metrizable space

Suppose $X$ is a completely metrizable (but not separable) space. Suppose $D$ is a Borel (actually $F_{\sigma}$) subset of $X$. Is there any logical relation between the following statements? [1] ...
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### Is there a suitably generalized Baire property for topological spaces of arbitrary cardinalities?

Is there some suitable generalization to the notion of Baire property for topological spaces of arbitrary cardinalities which satisfies the following condition: The meager sets are sets which are ...
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### Generalizations of the Tietze extension theorem (and Lusin's theorem)

I am reasking a year-old math.stackexchange.com question asked by someone else. (For my needs every space $X$ and $Y$ will be Polish---that is a completely separably metrizable space.) The Tietze ...
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Suppose $S\subset\mathbb{R}$ is dense without interior point, and for every open interval $I,J\subset\mathbb{R}$, $I\cap S$ is homeomorphic to $J\cap S$. Is $S\times S$ homeomorphic to $S$? By Luzin ...
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### Restrictions of null/meager ideal

Let I denote the null (resp. meager) ideal on reals. Is it consistent that for any pair of non null (resp. meager) sets A and B, there is a null (resp. meager) preserving bijection between A and B? In ...
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### Connectedness of the complement of small subsets (extended question)

The following questions occurred to me while browsing this site and looking at Exercise 20 here. Question 1. Let $n>1$. Does there exist a countable dense subset $A\subset\mathbb{R}^n$ for which ...
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### Descriptive set theory on $\mathbb{R}^\mathbb{N}$

The short version of my question is, What is a good source for learning about descriptive set theory on the space $\mathbb{R}^\mathbb{N}$, under the product topology coming from the discrete topology ...
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### Is the generalized Baire space complete?

I want to see whether the fact that the Baire space $\omega^\omega$ is a complete (metrizable) space generalizes to $\kappa^\kappa$ being a complete (topological) space. I think this is an easy ...
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### Is every $\sigma$-algebra of sets *abstractly* the Borel algebra of a topology on perhps some other set?

Is every sigma-algebra the Borel algebra of a topology? inspires the present question which asks for less. Question: Given a $\sigma$-algebra ${\cal A}$ on a set $X$, does there exist a topology ...
Let A=$\{a_n : n\in \omega \}\subset 2^{\omega\times\omega}$ be nonempty countable without isolated points (i.e. homeomorphic to $\mathbb{Q}$), and satisfy $\forall n\in \omega \exists^\infty ... 0answers 309 views ### Closure properties of familes of$G_\delta$sets. Given a family of sets$G\subset P(X)$, can one characterize by "closure properties" alone whether or not$G$arises as the family of all$G_\delta$for some topology on$X$? some Polish space ... 2answers 3k views ### Is every sigma-algebra the Borel algebra of a topology? This question arises from the excellent question posed on math.SE by Salvo Tringali, namely, Correspondence between Borel algebras and topology. Since the question was not answered there after some ... 1answer 313 views ### Action on a compact group If$G$is an infinite compact group, how many orbits can$G$have under the group action of its continuous automorphisms ? 2answers 498 views ### Question about 0-dimensional Polish spaces Hello everybody, I'm stuck with proving (or disproving) the following statement. Statement: For every$0$-dimensional Polish space$(X,\mathcal{T}\ )$, and a countable basis of clopen sets ... 4answers 569 views ### Continuously selecting elements from unordered pairs The symmetric square of a topological space$X$is obtained from the usual square$X^2$by identifying pairs of symmetric points$(x_1,x_2)$and$(x_2,x_1)$. Thus, elements of the symmetric square can ... 2answers 231 views ### When can the one-one continuous image of a perfect set fail to be perfect? Let$\mathfrak{M}$and$\mathfrak{N}$be perfect Polish spaces,$P$a nonempty perfect subset of$\mathfrak{M}$, and$f: \mathfrak{M} \rightarrow \mathfrak{N}$a continuous surjection that's injective ... 0answers 258 views ### For METRIZABLE spaces, do the Banach classes and Baire classes coincide? In this paper: 'Borel structures for Function spaces' by Robert Aumann, http://projecteuclid.org/euclid.ijm/1255631584 Aumann claims that when X and Y are metric spaces (among other things), the ... 3answers 594 views ### A compactness property for Borel sets Is the following generalised compactness property of Borel sets in a Polish space consistent with ZFC? (*) Let$\mathcal{B}$be a family of$\aleph_1$-many Borel sets. If$\bigcap \mathcal{B} = ...
Let $X$ be a topological space. When I call a set nowhere dense, meagre or similar without qualification, I mean that it has this property as a subset of $X$. Call a subset of $X$ weager (for weakly ...