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0 answers
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Is there a continuous map $f:\mathbb R^\omega\to\mathbb R^\omega$ with dense countable preimage $f^{-1}(\mathbb Q^\omega)$?

Let $\mathbb Q^\omega_0:=\{(x_i)_{i\in\omega}\in\mathbb Q^\omega:\exists n\in\omega\;\forall m\ge n\;\;x_m=0\}$ and observe that $\mathbb Q^\omega_0$ is a countable dense set in $\mathbb R^\omega$ (...
Taras Banakh's user avatar
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14 votes
0 answers
427 views

Which functions have all the common $\forall\exists$-properties of continuous functions?

This is an attempt at partial progress towards this question. Meanwhile, Sam Sanders pointed out that my original term was already in use, as were a couple other back-up terms, so ... oh well. For a ...
Noah Schweber's user avatar
12 votes
0 answers
172 views

A connected Borel subgroup of the plane

It is known that the complex plane $\mathbb C$ contain dense connected (additive) subgroups with dense complement but each dense path-connected subgroup of $\mathbb C$ necessarily coincides with $\...
Taras Banakh's user avatar
  • 41.9k
12 votes
0 answers
373 views

Does each compact topological group admit a discontinuous homomorphism to a Polish group?

A compact topological group $G$ is called Van der Waerden if each homomorphism $h:G\to K$ to a compact topological group is continuous. By a classical result of Van der Waerden (1933) the groups $SO(...
Taras Banakh's user avatar
  • 41.9k
11 votes
0 answers
144 views

Characterizing compact Hausdorff spaces whose all subsets are Borel

I am interested in characterizing compact topological spaces all of whose subsets are Borel. In this respect I have the following Conjecture. For a compact Hausdorff space $X$ the following ...
Taras Banakh's user avatar
  • 41.9k
10 votes
0 answers
323 views

Determinacy coincidence at $\omega_1$: is CH needed?

This is a follow-up to the last part of an old MSE answer of mine. Briefly, an analogue at $\omega_1$ of Steel's equivalence between clopen and open determinacy can be proved assuming $\mathsf{CH}$, ...
Noah Schweber's user avatar
10 votes
0 answers
272 views

What is the smallest $\sigma$-algebra of reals that is closed under addition of sets?

What is the smallest $\sigma$-algebra $\Sigma\subseteq\mathcal P(\Bbb R)$ containing the open sets and such that if $A,B\in\Sigma$, then $$A+B=\{a+b\mid a\in A,b\in B\}\in\Sigma?$$ I know that neither ...
Alessandro Codenotti's user avatar
10 votes
0 answers
498 views

Is there a model of set theory in which $\mathfrak p< \mathfrak b < \mathfrak q$?

Is there a model of set theory in which $\mathfrak p< \mathfrak b < \mathfrak q$? Here $\mathfrak p$, $\mathfrak b$, $\mathfrak q$ are small uncountable cardinals: $\mathfrak p$ is the ...
Alexander Osipov's user avatar
8 votes
0 answers
463 views

When is the sigma-algebra generated by closed convex sets the same as the Borel sigma-algebra

For which topological vector spaces $E$ do we have the equality between the sigma algebra generated by the closed convex subsets of $E$ and the Borel sigma algebra of $E$ ? More precisely, do we have ...
LCO's user avatar
  • 506
7 votes
0 answers
172 views

Countable network vs countable Borel network

Definition. A family $\mathcal N$ of subsets of a topological space $X$ is called $\bullet$ a network if for any open set $U\subset X$ and point $x\in U$ there exists a set $N\in\mathcal N$ such that $...
Taras Banakh's user avatar
  • 41.9k
7 votes
0 answers
466 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 ...
David Feldman's user avatar
6 votes
0 answers
255 views

Every Polish space is the image of the Baire space by a continuous and closed map, reference

The following result was originally proven by Engelking in his 1969 paper On closed images of the space of irrationals (AMS, JSTOR, MR239571, Zbl 0177.25501) Every Polish space (i.e. every separable ...
Lorenzo's user avatar
  • 2,286
6 votes
0 answers
210 views

Failure of Baire's grand theorem when the hypothesis is weakened to separable metric space

The statement of Baire grand theorem gives a characterization of Baire class 1 functions between a completely metrizable separable space (aka Polish space) and a separable metrizable space. The ...
Lorenzo's user avatar
  • 2,286
6 votes
0 answers
180 views

The Fubini property of the $\sigma$-ideal generated by closed subsets of measure zero

Question. Can a Borel subset $B\subset\mathbb R\times \mathbb R$ be covered by countably many closed sets of measure zero in the plane if for every $(a,b)\in\mathbb R\times\mathbb R$ the horisontal ...
Taras Banakh's user avatar
  • 41.9k
5 votes
0 answers
170 views

Can maximal filters of nowhere meager subsets of Cantor space be countably complete?

Let $X$ denote Cantor space. A subset $A\subseteq X$ is nowhere meager if for every non-empty open $U\subseteq X$, we have $A\cap U$ non-meager. We call $\mathcal{F}\subseteq \mathcal{P}(X)$ a maximal ...
Andy's user avatar
  • 369
5 votes
0 answers
113 views

Stronger form of countable dense homogeneity

I am completing my undergrad thesis about topological properties of some subspaces of the real numbers, and CDH spaces are one of the topics I´ve covered (I know almost nothing about it, I only prove ...
Saúl RM's user avatar
  • 10.6k
5 votes
0 answers
154 views

Is there a Hausdorff space whose "covering problem" has intermediate complexity?

For a "reasonable" pointclass ${\bf \Gamma}$, say that a second-countable space $(X,\tau)$ is ${\bf \Gamma}$-describable iff for some (equivalently, every) enumerated subbase $B=(B_i)_{i\in\...
Noah Schweber's user avatar
5 votes
0 answers
214 views

On generically Haar-null sets in the real line

First some definitions. For a Polish space $X$ by $P(X)$ we denote the space of all $\sigma$-additive Borel probability measures on $X$. The space $P(X)$ carries a Polish topology generated by the ...
Taras Banakh's user avatar
  • 41.9k
5 votes
0 answers
138 views

Disjoint covering number of an ideal

Let $\mathcal I$ be a $\sigma$-ideal with Borel base on an uncountable Polish space $X=\bigcup\mathcal I$. Let $\mathrm{cov}(\mathcal I)$ (resp. $\mathrm{cov}_\sqcup(\mathcal I)$) be the smallest ...
Taras Banakh's user avatar
  • 41.9k
4 votes
0 answers
127 views

An uncountable Baire γ-space without an isolated point exists?

An open cover $U$ of a space $X$ is: • an $\omega$-cover if $X$ does not belong to $U$ and every finite subset of $X$ is contained in a member of $U$. • a $\gamma$-cover if it is infinite and each $x\...
Alexander Osipov's user avatar
4 votes
0 answers
136 views

Is there a condensation of a closed subset of $\kappa^\omega$ onto $\kappa^\omega\setminus A$ …?

Let $\aleph_1\le\kappa<c$ and $A\subset \kappa^{\omega}$ such that $\lvert A\rvert\le\kappa$. Is there a condensation (i.e. a bijective continuous mapping) of a closed subset of $\kappa^\omega$ ...
Alexander Osipov's user avatar
4 votes
0 answers
140 views

Separable metrizable spaces far from being completely metrizable

I came across a kind of separable metrizable space that is "far" from being completely metrizable. Before specifying what I mean with "far", I recall that a space is said to be ...
Lorenzo's user avatar
  • 2,286
4 votes
0 answers
273 views

Sierpinski's characterization of $F_{\sigma\delta}$ spaces

According to [2]: Let $X$ be a space. We call a system $(X_s)_{s\in T}$ a Sierpinski stratification of $X$ if $T$ is a nonempty tree over a countable alphabet and $X_s$ is a closed subset of $X$ for ...
D.S. Lipham's user avatar
  • 3,317
4 votes
0 answers
64 views

Borel rank collapse in Hilbert cube modulo $\sigma$-ideal generated by zero-dimensional sets

Both of the commonly studied $\sigma$-ideals (meager sets and null sets) in Polish spaces with a natural measure (i.e. $\mathbb{R}$, $[0,1]$, $[0,1]^\omega$, $2^{\omega}$, etc.) have the nice property ...
James E Hanson's user avatar
4 votes
0 answers
105 views

Borel selections of usco maps on metrizable compacta

The problem posed below is motivated by this problem of Chris Heunen and in fact is its reformulation in the language of usco maps. Let us recal that an usco map is an upper semicontinuous compact-...
Taras Banakh's user avatar
  • 41.9k
4 votes
0 answers
122 views

Completely I-non-measurable unions in Polish spaces

Problem. Let $X$ be a Polish space, $\mathcal I$ be a $\sigma$-ideal with Borel base, and $\mathcal A\subset\mathcal I$ be a point-finite cover of $X$. Is it true that $\mathcal A$ conatins a ...
Lviv Scottish Book's user avatar
4 votes
0 answers
195 views

A kind of 0-1 law?

Suppose that P is a Borel subset of Baire $\times$ Baire, such that for every pair $x,x'$ of reals in the horizontal copy of Baire, if: $x,x'$ are $E_0$-equivalent (that is, $x(n)=y(n)$ for all but ...
Vladimir Kanovei's user avatar
4 votes
0 answers
229 views

Do $G_\delta$-measurable maps preserve dimension?

This question (in a bit different form) I leaned from Olena Karlova. Question. Let $f:X\to Y$ be a bijective continuous map between metrizable separable spaces such that for every open set $U\subset ...
Taras Banakh's user avatar
  • 41.9k
3 votes
0 answers
77 views

Can we generalize the Kuratowski Extension Theorem to Souslin spaces?

The Kuratowski Extension Theorem says: Let $(X,\mathcal{A})$ be a measurable space, $Y$ be a polish space, $A\subseteq X$, and $f:A\to Y$ be a measurable map. Then there is a measurable function $F:X\...
rfloc's user avatar
  • 637
3 votes
0 answers
141 views

Which cardinal $\kappa\geq \omega_1$ is critical for the following property...?

Which cardinal $\kappa\geq \omega_1$ is critical for the following property: Let $X\subset \mathbb R$ and $\kappa>|X|\geq \omega_1$. Then there is an uncountable family $\{X_{\alpha}\}$ such that $...
Alexander Osipov's user avatar
3 votes
0 answers
79 views

Is $\mathfrak q_0$ equal to the smallest cardinality of a second-countable $T_1$-space which is not a $Q$-space?

A topological space $X$ is a $Q$-space if every subset of $X$ is of type $G_\delta$. The smallest cardinality of a metrizable separable space which is not a $Q$-space is denoted by $\mathfrak q_0$ and ...
Taras Banakh's user avatar
  • 41.9k
3 votes
0 answers
77 views

What is the name of the (possibly well-known) class of $\pi$-monolithic compact spaces?

A compact space $X$ is called ${\it \pi-monolithic}$ if whenever a surjective continuous mapping $f:X\rightarrow K$ where $K$ is a compact metric space there exists a compact metric space $T\subseteq ...
Alexander Osipov's user avatar
3 votes
0 answers
80 views

Every Borel linearly independent set has Borel linear hull (reference?)

I am looking for a reference to the following fact, which probably is known and could be proved somewhere by someone. Theorem. The linear hull of any linearly independent Borel set in a Polish ...
Taras Banakh's user avatar
  • 41.9k
3 votes
0 answers
72 views

Borel complexity of special unions of Polish spaces

Let $X$ be a compact metrizable space and $(A_q)_{q\in\mathbb Q}$ be a family of pairwise disjoint sets, indexed by rational numbers. Assume that the family $(A_q)_{q\in\mathbb Q}$ has the following ...
Taras Banakh's user avatar
  • 41.9k
3 votes
0 answers
209 views

Nowhere Baire spaces

Studying the article "Barely Baire spaces" of W. Fleissner and K. Kunen, using stationary sets, they show an example of a Baire space whose square is nowhere Baire (we call a space $X$ nowhere Baire ...
Gabriel Medina's user avatar
3 votes
0 answers
92 views

Is there a T3½ category analogue of the density topology?

Motivation: I understand that various attempts have been made at defining a topology on $\mathbb{R}$ that is an analogue of the density topology ([1]) but for category (and meager sets) instead of ...
Gro-Tsen's user avatar
  • 32.5k
3 votes
0 answers
161 views

A characterization of Cauchy filters on countable metric spaces?

Given a filter $\mathcal F$ on a countable set $X$, consider the family $$\mathcal F^+:=\{A\subset X:\forall F\in\mathcal F\;(A\cap F\ne\emptyset)\}.$$ The following characterization is well-known. ...
Taras Banakh's user avatar
  • 41.9k
3 votes
0 answers
143 views

Is an Abelian topological group compact if it is complete and Bohr-compact?

A topological group $G$ will be called Bohr-compact if its Bohr topology (i.e., the largest precompact group topology) is compact and Hausdorff. A topological group $G$ is Bohr-compact if it admits ...
Taras Banakh's user avatar
  • 41.9k
3 votes
0 answers
207 views

Is the homeomorphism group of a Polish space a measurable group?

Let $X$ be a Polish space. Let $H(X)$ be the set of homeomorphisms $h \colon X \to X$, equipped with the "evaluation $\sigma$-algebra", namely $\sigma(h \mapsto h(x) : x \in X)$. (Note that for any ...
Julian Newman's user avatar
3 votes
0 answers
277 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 ...
Mario Carrasco's user avatar
2 votes
0 answers
49 views

$\sigma$-compactness of probability measures under a refined topology

Denote Polish spaces $(X, \tau_x)$ and $(Y, \tau_y)$, where $X$ and $Y$ are closed subsets of $\mathbb{R}$. Consider a Borel measurable function $f: (X \times Y, \tau_x \times \tau_y) \rightarrow \...
Hans's user avatar
  • 195
2 votes
0 answers
171 views

Is there a Lusin space $X$ such that ...?

Is there a Lusin space (in the sense Kunen) $X$ such that $X$ is Tychonoff; $X$ is a $\gamma$-space ? Note that if $X$ is metrizable and a $\gamma$-space then it is not Lusin. In mathematics, a ...
Alexander Osipov's user avatar
2 votes
0 answers
73 views

Separately continuous functions of the first Baire class

Problem. Let $X,Y$ be (completely regular) topological spaces such that every separately continuous functions $f:X^2\to\mathbb R$ and $g:Y^2\to \mathbb R$ are of the first Baire class. Is every ...
Lviv Scottish Book's user avatar
2 votes
0 answers
200 views

A question about infinite product of Baire and meager spaces

Proposition 1: For any space $X$ and an infinite cardinal $\kappa$, the product $X^{\kappa}$ is either meager or a Baire space. Does anyone have any suggestions to demonstrate Proposition 1? I was ...
Gabriel Medina's user avatar
2 votes
0 answers
101 views

A Baire space with meager projections

Question. Is there a Baire subspace $X$ of a Tychonoff power $M^\kappa$ of some separable metrizable space $M$ such that for any countable subset $A\subset \kappa$ the projection $$X_A=\{x{\...
Taras Banakh's user avatar
  • 41.9k
2 votes
0 answers
102 views

Is this concrete set generically Haar-null?

This question is related to this problem on MO about generically Haar-null sets in locally compact Polish groups but is more concrete. First we recall the definition of a generically Haar-null set in ...
Taras Banakh's user avatar
  • 41.9k
2 votes
0 answers
169 views

Classify spaces that make extension theorems hold

Recall a Polish space is a completely metrizable separable space. Say a Polish space $Y$ is a terminal space if for any Polish space $X$ and any closed $C \subseteq X$, one can extend a continuous ...
Jason Rute's user avatar
  • 6,287
2 votes
0 answers
371 views

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 ...
Noah Schweber's user avatar
1 vote
0 answers
83 views

Approximating evalutation maps at open sets over invariant measures

Let $G$ be a group acting by homeomorphisms on a compact metrizable space, say $X$; let's denote by $\alpha:G\to\mathrm{Homeo}(X)$ the action, $g\mapsto\alpha_g$, and consider the weak-$^*$ compact ...
J G's user avatar
  • 93
1 vote
0 answers
155 views

Study of the class of functions satisfying null-IVP

$\mathcal{N}_u$ : Class of all uncountable Lebesgue-null set i.e all uncountable sets having Lebesgue outer measure $0$. Let $f:\Bbb{R}\to \Bbb{R}$ be a function with the following property : $\...
SoG's user avatar
  • 307