All Questions
Tagged with gn.general-topology descriptive-set-theory
186
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
26
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 ...
- 4,096
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?
- 271
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
19
votes
1
answer
545
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 ...
- 16.5k
19
votes
1
answer
448
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 ...
- 40.2k
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
17
votes
1
answer
756
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?
- 40.2k
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
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
16
votes
1
answer
502
views
Does a completely metrizable space admit a compatible metric where all intersections of nested closed balls are non-empty?
(cross-posted from this math.SE question)
It is well-known that given a metric space $(X,d)$, the metric is complete if and only if every intersection of nested (i.e. decreasing with respect to ...
- 665
16
votes
1
answer
667
views
Question about product topology
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 ...
- 163
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
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
14
votes
2
answers
391
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,...
user44143
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
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
13
votes
1
answer
1k
views
Is every $\sigma$-algebra of sets *abstractly* the Borel algebra of a topology on perhaps 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 $\mathcal A$ on a set $X$, does there exist a topology $\...
- 17.2k
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
12
votes
2
answers
583
views
Partition $\Bbb{R}$ into a family of sets each one homeomorphic to the Cantor set
It is known that there is no (nontrivial) partition of $\Bbb{R}$ into a countable number of closed set. But is there a partition of $\Bbb{R}$ into sets, each one homeomorphic to the cantor ternary set?...
user36763
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
12
votes
1
answer
309
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 ...
- 40.2k
12
votes
0
answers
171
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 $\...
- 40.2k
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
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
11
votes
2
answers
973
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 ...
- 967
11
votes
1
answer
519
views
Non meager rectangle
Suppose $G \subseteq \mathbb{R}^2$ is dense $G_\delta$. Must there (in ZFC) exist non meager sets of reals $A, B$ such that $A \times B \subseteq G$?
- 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
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
10
votes
3
answers
838
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} = \...
- 916
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
10
votes
1
answer
352
views
Two dimensional perfect sets
Consider the following family of sets
$$ \begin{align*}
\mathcal{F} = \{X\subseteq [0,1]\times [0,1] \mid \ &X \text{ is closed and }\\& \forall x \in \pi_0 (X) (\{y \in [0,1] \mid (x,y) \in ...
- 2,042
10
votes
2
answers
342
views
Source on smooth equivalence relations under continuous reducibility?
This question was asked and bountied at MSE, but received no answer.
In the context of Borel reducibility, smooth equivalence relations (see the introduction of this paper) are rather boring since ...
- 19k
10
votes
0
answers
309
views
Determinacy coincidence at $\omega_1$: is CH needed?
This is a follow-up to the last part of an old MSE answer of mine. Briefly, an analogue at $\omega_1$ of Steel's equivalence between clopen and open determinacy can be proved assuming $\mathsf{CH}$, ...
- 19k
10
votes
0
answers
248
views
What is the smallest $\sigma$-algebra of reals that is closed under addition of sets?
What is the smallest $\sigma$-algebra $\Sigma\subseteq\mathcal P(\Bbb R)$ containing the open sets and such that if $A,B\in\Sigma$, then $$A+B=\{a+b\mid a\in A,b\in B\}\in\Sigma?$$
I know that neither ...
- 2,672
10
votes
0
answers
485
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 ...
- 977
9
votes
2
answers
474
views
Can you fit a $G_\delta$ set between these two sets?
Every subset of $\mathbb N \times \mathbb N$ can be viewed as a relation on $\mathbb N$. The set $\mathcal P(\mathbb N \times \mathbb N)$ of all relations on $\mathbb N$ has a natural topology with ...
- 17.1k
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
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
9
votes
1
answer
557
views
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 ...
- 2,971
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
8
votes
2
answers
2k
views
Wanted: chain of nowhere dense subsets of the real line whose union is nonmeagre, or even contains intervals
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 ...
- 672
8
votes
1
answer
716
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....
- 155
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
193
views
Can totally inhomogeneous sets of reals coexist with determinacy?
A special case of a theorem of Brian Scott (from On the existence of totally inhomogeneous spaces) is that there is a size-continuum set $S\subset\mathbb{R}$ such that if $x,y\in S$ are distinct then $...
- 19k
8
votes
1
answer
328
views
"Compactness length" of Baire space
Intuitively, my question is: how many times do we have to mod out by an closed equivalence relation with all classes compact in order to collapse Baire space $\omega^\omega$ to a singleton?
In more ...
- 19k
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
8
votes
2
answers
703
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 ...
- 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