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Base zero-dimensional spaces

Definition. A zero-dimensional topological space $X$ is called base zero-dimensional if for any base $\mathcal B$ of the topology that consists of closed-and-open sets in $X$, any open cover $\mathcal ...
Taras Banakh's user avatar
  • 41.8k
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
7 votes
1 answer
296 views

Can we inductively define Wadge-well-foundedness?

For a topological space $X$ (which I'll identify with its underlying set of points), we define the Wadge preorder $Wadge(X)$: elements of the preorder are subsets of $X$, and the ordering is given by $...
Noah Schweber's user avatar
9 votes
1 answer
336 views

How much can complexities of bases of a "simple" space vary?

Given a countable subbase of a topology, we can consider its complexity in terms of the difficulty of determining whether one family of basic open sets covers another basic open set. My question is ...
Noah Schweber's user avatar
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
  • 41.8k
8 votes
2 answers
1k views

When does an "$\mathbb{R}$-generated" space have a short description?

The following is a more focused version of the original question; see the edit history if interested. In the original version of the question, five other variants of the "simplicity" ...
Noah Schweber's user avatar
13 votes
1 answer
519 views

When can I "draw" a topology in Baire space?

The motivation for this question is a bit convoluted, so in the interests of conciseness I'm just asking it as a curiosity (and I do find it interesting on its own); if anyone is interested, feel free ...
Noah Schweber's user avatar
2 votes
1 answer
133 views

Topologically Ordered Families of Disjoint Cantor Sets in $I$?

Suppose that we have an uncountable collection $C_\alpha$ of disjoint Cantor Sets contained in the closed unit interval $I$. Suppose we have ordered the indices $\alpha \in [0,1]$ as well. Then is ...
John Samples's user avatar
19 votes
1 answer
556 views

Can an injective $f: \Bbb{R}^m \to \Bbb{R}^n$ have a closed graph for $m>n$?

Question. Suppose $m>n$ are positive integers. Is there a one-to-one $f: \Bbb{R}^m \to \Bbb{R}^n$ such that the graph $\Gamma_f$ of $f$ is closed in $\Bbb{R}^{m+n}$? Remark 1. The answer to the ...
Ali Enayat's user avatar
  • 17.7k
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
  • 41.8k
27 votes
1 answer
4k views

Closed balls vs closure of open balls

We work in a separable metric space $(X,d)$. With $\overline{B}(x,r)$ I denote the closed ball around $x$ of radius $r$, and with $cl \ B(x,r)$ I denote the closure of the open ball. Clearly, we ...
Arno's user avatar
  • 4,727
3 votes
1 answer
119 views

Nice representation of open sets in $\sigma$-algebras in certain circumstances

Let $(X,\tau)$ be a topological space. For a given topological base $\mathcal{E}$ for $\tau$, let us denote Bor$(\mathcal{E})$, by the smallest $\sigma$-algebra containing $\mathcal{E}$. Q. Assume ...
ABB's user avatar
  • 4,058
4 votes
1 answer
121 views

Nice arrangement of open sets in $\sigma$-algebras

Let $X$ be a topological space and $\mathcal{E}$ be a topological base for $X$. Let us denote Bor$(\mathcal{E})$, by the smallest $\sigma$-algebra containing $\mathcal{E}$. Q. Let $O$ be an open ...
ABB's user avatar
  • 4,058
1 vote
0 answers
93 views

An example of a Borel map of the first class

Let $X,Y$ be compact metric spaces, $2^X$ the set of all closed subsets of $X$ and $f:X\to Y$ be a 1st class Borel mapping. Im trying to check Borel class of mapping $G:2^Y\to 2^X$. I submit it in a ...
Tony T.'s user avatar
  • 21
3 votes
1 answer
285 views

Example of a Baire Class $1$ function $f$ satisfying $\omega\cdot n<\beta(f)\leq \omega\cdot (n+1)$ for some natural number $n\geq 1.$

Definitions: Let $X$ be a Polish space (separable completely metrizable topological space). A function $f:X\to\mathbb{R}$ is Baire Class $1$ if it is a pointwlise limit of a sequence of continuous ...
Idonknow's user avatar
  • 623
11 votes
1 answer
704 views

Examples of Baire Class $\xi+1$ but not $\xi$ functions for each countable ordinal $\xi.$

We say that $f:\mathbb{R}\to\mathbb{R}$ is of Baire Class $1$ if it is a pointwise limit of a sequence of continuous functions. One can generalize the definition above by taking pointwise limit of ...
Idonknow's user avatar
  • 623
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
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
7 votes
1 answer
288 views

Is there a first-countable space containing a closed discrete subset which is not $G_\delta$?

Being motivated by this problem, I am searching for an example of a first-countable regular topological space $X$ containing a closed discrete subset $D$, which is not $G_\delta$ in $X$. It is easy ...
Taras Banakh's user avatar
  • 41.8k
12 votes
1 answer
582 views

Is a locally finite union of $G_\delta$-sets a $G_\delta$-set?

Problem. Let $\mathcal F$ be a locally finite (or even discrete) family of (closed) $G_\delta$-sets in a topological space $X$. Is the union $\cup\mathcal F$ a $G_\delta$-set in $X$? Remark. The ...
Taras Banakh's user avatar
  • 41.8k
5 votes
1 answer
375 views

Equivalent of Lusin's Theorem in Borel setting

Let $X$ be a Polish space, $\mathcal B$ the sigma-algebra of Borel sets. Let $E$ be an aperiodic countable Borel equivalence relation on $X \times X$ (this means that every class of equivalence ...
Shrey's user avatar
  • 53
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
  • 41.8k
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
  • 41.8k
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
  • 41.8k
3 votes
1 answer
468 views

If non-empty player has a winning strategy in Banach-Mazur game BM(X), then it also has in BM(Y)?

Let $f:X\rightarrow Y$ be a continuous, open, surjection function and second player (non-empty) has a winning strategy (not important which one, say for simplicity stationery st.) in $BM(X)$. Then can ...
user117537's user avatar
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
  • 41.8k
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
  • 41.8k
8 votes
2 answers
759 views

A representation of $F_{\sigma\delta}$-ideals?

First some definitions. By $\mathcal P(\mathbb N)$ we denote the family of all subsets of $\mathbb N$ endowed with the metrizable separable topology generated by the countable base consisting of the ...
Taras Banakh's user avatar
  • 41.8k
15 votes
0 answers
409 views

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
  • 41.8k
17 votes
1 answer
794 views

Is there a continuous function $f:\mathbb R^\omega\to\mathbb R$ with injective restriction $f|\mathbb Q^\omega$?

Question. Is there a continuous function $f:\mathbb R^\omega\to\mathbb R$ whose restriction $f|\mathbb Q^\omega$ is injective?
Taras Banakh's user avatar
  • 41.8k
7 votes
1 answer
209 views

Are σ-sets preserved by Borel isomorphisms?

Recall that a $\sigma$-space is a topological space such that every $F_{\sigma}$-set is $G_{\delta}$-set. $X$ - $\sigma$-set, if $X$ is a $\sigma$-space and it is subset of real line $R$. Let $F$ ...
Alexander Osipov's user avatar
3 votes
4 answers
653 views

Picking a real for every non-empty open set in $\mathbb{R}$

Let ${\cal E}$ denote the collection of open sets of $\mathbb{R}$ with respect to the Euclidean topology. It is well known that $|{\cal E}| = 2^{\aleph_0}$. Is there an injective map $f:{\cal E}\...
Dominic van der Zypen's user avatar
7 votes
1 answer
374 views

Is each $G_\delta$-measurable map $\sigma$-continuous?

Definition. A function $f:X\to Y$ between topological spaces is called $\bullet$ $G_\delta$-measurable if for each open set $U\subset Y$ the preimage $f^{-1}(U)$ is of type $G_\delta$ in $X$; $\...
Taras Banakh's user avatar
  • 41.8k
12 votes
1 answer
316 views

A reference to a theorem on the equivalence of ideals of measure zero in the Cantor cube

I am looking for a reference of the following (true) fact: Theorem. For any two continuous strictly positive Borel probability measures $\mu,\lambda$ on the Cantor cube $2^\omega$ there exists a ...
Taras Banakh's user avatar
  • 41.8k
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
  • 41.8k
12 votes
0 answers
372 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
  • 41.8k
4 votes
1 answer
348 views

Is there a topologizable group admitting only Raikov-complete group topologies?

Definition. A group $G$ is called complete (resp. non-topologizable) if each Hausdorff group topology on $G$ is Raikov-complete (resp. discrete). It is clear that each non-topologizable group is ...
Taras Banakh's user avatar
  • 41.8k
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
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
1 vote
1 answer
245 views

Definition of $F_{\sigma}$ sets in terms of $\varepsilon$?

Let $X$ be a metric space. In Borel hierarchy, $\Sigma_{1}^0$ is the set of all open sets in $X$ while $\Pi_{1}^0$ is the set of all closed sets in $X.$ Then at next level, one has $\Sigma_{2}^0 = \{...
Idonknow's user avatar
  • 623
19 votes
1 answer
465 views

Large Borel antichains in the Cantor cube?

Let $2^\omega$ be the Cantor cube $\{0,1\}^\omega$, endowed with the standard compact metrizable topology and the standard product measure, called the Haar measure. The Cantor cube is considered as a ...
Taras Banakh's user avatar
  • 41.8k
14 votes
2 answers
413 views

Given a sequence of reals, we can find a dense sequence avoiding it, but can we find one continuously?

Let $S$ be the set of injective sequences in $\mathbb{R}$: $$S = \{s: \mathbb{N} \rightarrow \mathbb{R}: s(m) \neq s(n) \text{ if }m \neq n\}.$$ Consider $S$ with the topology of pointwise convergence,...
user avatar
4 votes
1 answer
470 views

Covering measure one sets by closed null sets

(The following question arose in a joint research with Adam Przeździecki and Boaz Tsaban.) For a $\sigma$-ideal $\mathcal{I}$ of subsets of the unit interval $[0,1]$, define $$\newcommand{\card}[1]{\...
Piotr Szewczak's user avatar
11 votes
2 answers
1k views

How to show that something is not completely metrizable

I have a Polish space $X$ and a subset $A \subset X$. I know that $A$ is completely metrizable (in its induced topology) if and only if $A$ is a $G_\delta$-set in $X$. This means: If I want to show ...
Tom's user avatar
  • 987
13 votes
3 answers
820 views

Is there a Borel subset of $ \mathbb{R}^{2} $, with finite vertical cross-sections, whose projection onto the first component is non-Borel?

This question is related to another one that I asked two days ago. Question. Does there exist a Borel subset $ M $ of $ \mathbb{R}^{2} $ with the following two properties? The ...
Transcendental's user avatar
6 votes
4 answers
2k views

A simpler proof that compact sets have cardinality continuum?

Is there a simple reason why uncountable compact sets of real numbers have cardinality continuum? I know that this is immediate from the Cantor-Bendixon Theorem, but I wonder whether this consequence ...
Boaz Tsaban's user avatar
  • 3,104
8 votes
1 answer
851 views

When the boundary of any subset is compact?

Let $X$ be a Tychonoff space with no isolated points such that the boundary of any subset of $X$ is compact. Does it mean that $X$ is compact ? (If $X$ is a resolvable space then it is clearly compact....
Lisa_K's user avatar
  • 155
6 votes
1 answer
248 views

Is the space of countable closed covers of the Cantor set analytic?

For an uncountable compact metric space $X$ denote by $K(X)$ be the hyperspace of non-empty compact subsets of $X$, endowed with the Vietoris topology (which is generated by the Hausdorff metric). ...
Taras Banakh's user avatar
  • 41.8k
7 votes
2 answers
352 views

Topological spaces with too many open sets

Is there a Tychonoff space $X$ without isolated points with the following property: For any $a\in X$ and any function $f : X\longrightarrow \mathbb{R}$, if $f$ is continuous on $X\backslash \{a\}$ ...
alex alexeq's user avatar
  • 1,881
25 votes
3 answers
2k views

A rare property of Hausdorff spaces

Is there a Hausdorff topological space $X$ such that for any continuous map $f: X\longrightarrow \mathbb{R}$ and any $x\in \mathbb{R}$, the set $f^{-1}(x)$ is either empty or infinite?
Mark 's user avatar
  • 271