All Questions
Tagged with gn.general-topology descriptive-set-theory
186
questions
6
votes
1
answer
324
views
A strong Borel selection theorem for equivalence relations
In Kechris' book "Classical Descriptive Set Theory" there is the following theorem (12.16):
Let $X$ be a Polish space and $E$ an equivalence relation such that every equivalence class is ...
- 355
5
votes
0
answers
167
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 ...
- 369
3
votes
1
answer
163
views
A Borel perfectly everywhere dominating family of functions
Is there a Borel function $f:2^\omega\to\omega^\omega$ such that for every nonempty closed perfect set $P\subseteq 2^\omega$, $f|P$ is a dominating family of functions in $\omega^\omega$?
This is a ...
- 2,971
1
vote
2
answers
239
views
A Borel perfectly everywhere surjective function on the Cantor set
Does there exist a Borel (or even continuous) function $f:\mathcal{C}\to\mathcal{C}$, where $\mathcal{C}$ is the Cantor set (or Cantor space $2^\omega$) such that for every nonempty closed perfect set ...
- 2,971
5
votes
1
answer
240
views
How complex is the orbit equivalence relation of $\mathrm{Iso}_0(X)\curvearrowright S_X$ for $X=L^p([0,1])$?
For a Banach space $X$ let $S_X$ denote its unit sphere and let $\mathrm{Iso}_0(X)$ denote the group of rotations of $X$, that is isometries fixing the origin. There is a natural continuous action $\...
- 2,672
5
votes
0
answers
112
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 ...
- 7,836
1
vote
1
answer
248
views
CH and the density topology on $\mathbb{R}$
In the article AN EXAMPLE INVOLVING BAIRE SPACES (https://www.ams.org/journals/proc/1975-048-01/S0002-9939-1975-0362249-1/S0002-9939-1975-0362249-1.pdf) of H. E. White Jr. it is shown that, assuming ...
- 551
1
vote
0
answers
152
views
$f:Y\to X$ continuous with $f^{-1}(x)$ compact for $x\in X$, does there exist a Borel measurable map $g:X\to Y$?
Let $X,Y$ be Polish, metric spaces. $f:Y\to X$ is a continuous, surjective map and for any $x\in X$, $f^{-1}(x)\subset Y$ is compact. Is it true that there is a injective, Borel measurable map $g:X \...
5
votes
0
answers
167
views
What is known about these "explicitly represented" spaces?
Apologies if this is too low-level. A related question that I asked on the Math Stack Exchange got no answers after a year, so I thought it might be better to ask this one here.
The standard approach ...
- 3,398
3
votes
0
answers
68
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 ...
- 40.2k
5
votes
0
answers
151
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\...
- 19k
4
votes
1
answer
670
views
Is every element of $\omega_1$ the rank of some Borel set?
It is well known that we can obtain the $\sigma$-algebra of Borel subsets of $2^{\omega}$ in the following way: Let $B_0$ be the collection of all open subsets of $2^{\omega}$. For $\alpha=\beta+1$, ...
- 1,110
4
votes
0
answers
261
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 ...
- 2,993
5
votes
1
answer
274
views
Is the Hilbert cube the countable union of punctiform spaces?
Recall that a (separable) metric space is called punctiform, if all its compact subspaces are zero-dimensional. While "natural" spaces would seem to be punctiform if they already themselves ...
- 4,096
2
votes
2
answers
125
views
Finding 1-generic paths through a tree $T \subseteq 2^{<\omega}$
Consider Cantor space $2^\omega$ with the standard topology generated by open sets $[\sigma] = \{ \sigma^\frown x: x \in 2^\omega \}$. If $A \subseteq 2^{<\omega}$ and $x \in 2^\omega$, we say $A$ ...
11
votes
2
answers
584
views
Homeomorphisms and "mod finite"
Suppose $f:C\to C$ is a homeomorphism, where $C=\{0,1\}^{\mathbb N}$ is Cantor space.
Suppose $f$ preserves $=^*$ (equality on all but finitely many coordinates). Does it follow that $f$ also reflects ...
- 24.5k
6
votes
1
answer
378
views
What is the Borel complexity of this set?
Problem. What is the Borel complexity of the set
$$c(\mathbb Q)=\{(x_n)_{n\in\omega}\in\mathbb R^\omega:\exists\lim_{n\to\infty}x_n\in\mathbb Q\}$$
in the countable product of lines $\mathbb R^\omega$?...
- 40.2k
16
votes
1
answer
2k
views
Topological proof that a Vitali set is not Borel
This question is purely out of curiosity, and well outside my field — apologies if there is a trivial answer. Recall that a Vitali set is a subset $V$ of $[0,1]$ such that the restriction to $V$ of ...
- 36.9k
1
vote
1
answer
219
views
Is there a simple proof that proves $C^1[0, 1]$ is $\Sigma^1_1$ in $C[0, 1]$?
In his book, "Descriptive Set Theory", Moschovakis states $C^1[0, 1]$ is $\boldsymbol{\Sigma}^1_1$ in $C[0, 1]$ in the exercise 1E.8.
Here, $C[0, 1]$ is the space (metrized by the sup norm) of ...
- 111
1
vote
1
answer
237
views
Borel hierarchy and tail sets
Let $A$ be a finite set, and let $A^\infty$ be the set of all sequences $(a_n)_{n=1}^\infty$ of elements of $A$.
A set $B \subseteq A^\infty$ is a tail set if for every two sequences $\vec a, \vec b \...
- 705
4
votes
0
answers
60
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 ...
- 10.1k
2
votes
1
answer
179
views
Detecting comprehension topologically
This question basically follows this earlier question of mine but shifting from standard systems of nonstandard models of $PA$ to $\omega$-models of $RCA_0$. For $X$ a Turing ideal we get the map $c_X$...
- 19k
22
votes
1
answer
714
views
Undetermined Banach-Mazur games in ZF?
This question was previously asked and bountied on MSE, with no response. This MO question is related, but is also unanswered and the comments do not appear to address this question.
Given a ...
- 19k
6
votes
1
answer
472
views
Is there an almost strongly zero-dimensional space which is not strongly zero-dimensional
A Tychonoff space $X$ is called strongly zero-dimensional if each functionally closed subset $F$ of $X$ is a $C$-set, which means that $F$ is the intersection of a sequences of clopen sets in $X$.
A ...
- 4,395
4
votes
2
answers
580
views
Question about additive subgroups of the real line and the density topology
I am new studying additive subgroups of the real line, I would like to know if someone could give me an idea for the next question.
Let $m$ be the Lebesgue measure in $\mathbb{R}$. A measurable set $E\...
- 551
10
votes
2
answers
342
views
Source on smooth equivalence relations under continuous reducibility?
This question was asked and bountied at MSE, but received no answer.
In the context of Borel reducibility, smooth equivalence relations (see the introduction of this paper) are rather boring since ...
- 19k
4
votes
1
answer
498
views
The Borel class of a countable union of $G_\delta$-sets, which are absolute $F_{\sigma\delta}$
Problem. Assume that a metrizable separable space $X$ is the countable union $X=\bigcup_{n\in\omega}X_n$ of pairwise disjoint $G_\delta$-sets $X_n$ in $X$ such that each $X_n$ is an absolute $F_{\...
- 40.2k
8
votes
1
answer
252
views
Do the higher levels of the Borel hierarchy correspond to absolute topological properties?
It is well known that a subset $Y$ of a Polish space $X$ is completely metrisable iff it is a $G_\delta$ subset. This relates a relative topological property of the subspace $Y \subset X$ to an ...
- 1,761
3
votes
0
answers
208
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 ...
- 551
3
votes
1
answer
209
views
Product of Bernstein sets
Remember that a Bernstein set is a set
$B\subseteq \mathbb{R}$ with the property that for any uncountable closed set, $S$, in the real line both
$B\cap S$ and $(\mathbb{R}\setminus B)\cap S$ are non-...
- 551
4
votes
1
answer
391
views
Is there a universally meager air space?
Let $\mathcal P$ be a family of nonempty subsets of a topological space $X$. A subset $D\subset X$ is called $\mathcal P$-generic if for any $P\in\mathcal P$ the intersection $P\cap D$ is not empty.
A ...
- 40.2k
4
votes
0
answers
103
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-...
- 40.2k
6
votes
2
answers
190
views
A non-Borel union of unit half-open squares
On the complex plane $\mathbb C$ consider the half-open square $$\square=\{z\in\mathbb C:0\le\Re(z)<1,\;0\le\Im(z)<1\}.$$
Observe that for every $z\in \mathbb C$ and $p\in\{0,1,2,3\}$ the set $...
- 40.2k
7
votes
2
answers
473
views
Do continuous maps factor through continuous surjections via Borel maps?
Let $f \colon X \twoheadrightarrow Y$ be a continuous surjection between compact Hausdorff spaces, and $g \colon \mathbb{R} \to Y$ a continuous function. Can you always find a Borel-measurable ...
- 3,909
8
votes
1
answer
389
views
Complexity of the set of closed subsets of an analytic set
Let $X$ be a compact Polish space and $K(X)$ the hyperspace of closed subspaces of $X$ with the Vietoris/Hausdorff metric topology.
Question: If $A$ is an analytic subset of $X$, what is the ...
- 2,971
3
votes
1
answer
139
views
A reference for a (folklore?) characterization of K-analytic spaces
I am writing a paper on K-analytic spaces and need the following known characterization.
Theorem. For a regular topological space $X$ the following conditions are equivalent:
(1) $X$ is a continuous ...
- 40.2k
4
votes
1
answer
217
views
K-analytic spaces whose any compact subset is countable
A regular topological space $X$ is called
$\bullet$ analytic if $X$ is a continuous image of a Polish space;
$\bullet$ $K$-analytic if $X$ is the image of a Polish space $P$ under an upper ...
- 40.2k
3
votes
0
answers
90
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 ...
- 28.7k
9
votes
2
answers
459
views
Small uncountable cardinals related to $\sigma$-continuity
A function $f:X\to Y$ is defined to be
$\sigma$-continuous (resp. $\bar \sigma$-continuous) if there exists a countable (closed) cover $\mathcal C$ of $X$ such that the restriction $f{\restriction}C$ ...
- 40.2k
0
votes
1
answer
147
views
Reference request: Baire class 2 functions
There are many articles on Baire 1 functions, but not many on Baire 2 and above. Where can I find a nice comprehensive survey of them?
- 2,029
2
votes
0
answers
99
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{\...
- 40.2k
8
votes
1
answer
557
views
Is a Borel image of a Polish space analytic?
A topological space $X$ is called analytic if it is a continuous image of a Polish space, i.e., the image of a Polish space $P$ under a continuous surjective map $f:P\to X$.
We say that a topological ...
- 40.2k
6
votes
0
answers
157
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 $...
- 40.2k
5
votes
1
answer
514
views
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 ...
- 40.2k
4
votes
0
answers
120
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 ...
- 4,395
7
votes
1
answer
279
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 $...
- 19k
8
votes
1
answer
329
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 ...
- 19k
11
votes
0
answers
141
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 ...
- 40.2k
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" ...
- 19k
13
votes
1
answer
500
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 ...
- 19k