All Questions
Tagged with gn.general-topology descriptive-set-theory
37
questions
91
votes
3
answers
13k
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 ...
- 220k
10
votes
1
answer
724
views
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 ...
- 1,436
5
votes
0
answers
223
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 ...
- 40.2k
13
votes
0
answers
382
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 $\mathbb R^\omega$ (...
- 40.2k
4
votes
1
answer
1k
views
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 $\sim_f\...
- 1,595
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
17
votes
6
answers
2k
views
The reals as continuous image of the irrationals
In the Wikipedia article about descriptive set theory I read that $\mathbb{R}$ (with its usual topology) is a Polish space, and that every Polish space
1) can be obtained as a continuous image of ...
- 22.4k
17
votes
4
answers
1k
views
Continuity on a measure one set versus measure one set of points of continuity
In short: If $f$ is continuous on a measure one set, is there a function $g=f$ a.e. such that a.e. point is a point of continuity of $g$?
Now more carefully, with some notation: Suppose $(X, d_X)$ ...
- 2,209
16
votes
4
answers
724
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 ...
- 43.5k
15
votes
2
answers
3k
views
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 ...
- 6,167
14
votes
0
answers
406
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 ...
- 19k
14
votes
3
answers
799
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 ...
- 1,386
12
votes
0
answers
365
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(...
- 40.2k
12
votes
1
answer
774
views
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 ...
- 9,781
11
votes
1
answer
555
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 ...
- 40.2k
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
9
votes
1
answer
642
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 ...
- 603
9
votes
1
answer
393
views
Meager subgroups of compact groups
Suppose we have an infinite compact (Hausdorff) group $G$, and a subgroup $H\leq G$ which is meagre.
Can $H$ always be covered by a countable family of nowhere dense sets $H_n$ such that $H_n^2$ is ...
- 1,132
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
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
7
votes
1
answer
277
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 ...
- 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
7
votes
1
answer
224
views
Does a continuous map from $\kappa^\omega$ to $[0,1]^\omega$ have a non-scattered fiber?
Question. Let $\kappa>\mathfrak c$ be a cardinal endowed with the discrete topology and $f:\kappa^\omega\to[0,1]^\omega$ be a continuous map. Is there a point $y\in[0,1]^\omega$ whose preimage $f^{-...
- 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
6
votes
1
answer
175
views
On continuous perturbations of functions of the first Baire class on the Cantor set
Is it true that for any function of the first Baire class $f:X\to\mathbb R$ on the Cantor cube $X=2^\omega$ there is a continuous function $g:X\to[0,1]$ such that the image $(f+g)(X)$ is disjoint with ...
- 40.2k
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
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
6
votes
1
answer
320
views
Bernstein sets of large cardinality
A metrizable space $X$ will be called a generalized Bernstein set if every closed completely metrizable subspace $C$ of $X$ has cardinality $|C|<|X|$.
It is well-known that the real line contains ...
- 40.2k
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
5
votes
0
answers
212
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 ...
- 40.2k
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
1
answer
315
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 ...
- 40.2k
3
votes
1
answer
458
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 ...
- 31
3
votes
1
answer
159
views
Co-analytic $Q$-sets
A subset $A\subseteq \mathbb{R}$ is said to be a $Q$-set if every subset $B\subseteq A$ is $F_\sigma$ wrt the subspace topology on $A$. For example $\mathbb{Q}$ is a $Q$-set. The first time I have ...
- 2,042
3
votes
1
answer
1k
views
Different Metrics for Baire Space and their induced Topologies
The Baire-Space is the set of all infinite sequences of integers, i.e.
$$
\mathcal N = \omega^{\omega}.
$$
On this space usually the following metric is given
$$
d(\alpha, \beta) = \left\{ \begin{...
- 798
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
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