All Questions
Tagged with descriptive-set-theory gn.general-topology
193 questions
8
votes
1
answer
278
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 ...
- 3,612
8
votes
1
answer
261
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 ...
- 1,955
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" ...
- 21.2k
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 ...
- 506
7
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 ...
- 662
7
votes
1
answer
379
views
What is an example of a meager space X such that X is concentrated on countable dense set?
A topological space $X$ is concentrated on a set $D$ iff for any open set $G$ if $D\subseteq G$, then $X\setminus G$ is countable.
What is an example of a separable metrizable (uncountable) meager (...
- 1,101
7
votes
3
answers
356
views
Hausdorff quasi-Polish spaces
A topological space is said to be quasi-Polish if it is second-countable and completely quasi-metrizable (see for an introduction de Brecht's article: de Brecht, Matthew, Quasi-Polish spaces, Ann. ...
- 2,286
7
votes
2
answers
765
views
Question about 0-dimensional Polish spaces
Hello everybody,
I'm stuck with proving (or disproving) the following statement.
Statement:
For every $0$-dimensional Polish space $(X,\mathcal{T}\ )$, and a countable basis of clopen sets $\mathcal{...
user11618
7
votes
1
answer
185
views
Existence of Borel uniformization for coanalytic set with non-$K_\sigma$ sections
Suppose that $X$ is a Polish (or standard Borel) space and $\omega^\omega$ is the Baire space of all natural number sequences. My question is: If $A\subseteq X\times \omega^\omega$ is a coanalytic set ...
- 3,115
7
votes
1
answer
558
views
In a Polish space, is every analytic set the continuous image of a Borel set from the same Polish space?
I'm confused by a subtle point in the definition of analytic sets. Suppose I have a Polish space $X$. Now I start with the collection of Borel sets in $X$ and take all their continuous images in $X$. ...
- 113
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 ...
- 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 $X\backslash \{a\}$ ...
- 1,881
7
votes
2
answers
500
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 ...
- 3,937
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 $F$ ...
- 1,101
7
votes
1
answer
459
views
Product of limit $\sigma$-algebras
Let $X$ and $Y$ be Polish (i.e. Borel subsets of separable completely metrizable) spaces. For a Polish space $Z$, let $\mathscr{S}(Z)$ denote the limit $\sigma$-algebra on $Z$, i.e. the smallest $\...
- 275
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 $...
- 21.2k
7
votes
1
answer
236
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^{-...
- 41.8k
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 $X$;
$\...
- 41.8k
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 $...
- 41.8k
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 ...
- 17.6k
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 ...
- 3,104
6
votes
3
answers
1k
views
Borel cross section
It is known from metric space topology that a closed equivalence relation on a Polish space has either countably many or $\mathfrak{c}$ many equivalence classes.
A short elementary proof is given in ...
- 1,704
6
votes
1
answer
149
views
Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective?
Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective ?
- 1,101
6
votes
1
answer
353
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 ...
- 365
6
votes
1
answer
423
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 $\mathbb R^\omega$?...
- 41.8k
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).
...
- 41.8k
6
votes
1
answer
188
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 ...
- 41.8k
6
votes
2
answers
192
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 $...
- 41.8k
6
votes
1
answer
184
views
Classification of Polish spaces up to a $\sigma$-homeomorphism
A function $f:X\to Y$ between topological spaces is called
$\bullet$ $\sigma$-continuous if there exists a countable cover $\mathcal C$ of $X$ such that for every $C\in\mathcal C$ the restriction $f{\...
- 41.8k
6
votes
1
answer
489
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 ...
- 4,535
6
votes
1
answer
335
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 ...
- 41.8k
6
votes
0
answers
253
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 ...
- 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 ...
- 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 ...
- 41.8k
5
votes
2
answers
247
views
Definability properties of box-open subsets of Polish space
Let $X$ be a perfect Polish space $X$, so that $X^\omega$ is also a Polish space under the product topology. Call a subset $\mathcal{X} \subseteq X^\omega$ box-open if it is an open subset of $X^\...
- 1,382
5
votes
1
answer
370
views
Is it true that $\mathit{MA}(\omega_1)$ iff $\omega_1<\mathfrak{p}$?
Recall that
$\mathfrak{p}=\min\{|F|: F$ is a subfamily of $[\omega]^{\omega}$ with the sfip which has no infinite pseudo-intersection $\}$.
The cardinal $\mathfrak{q}_0$ defined as the smallest ...
- 1,101
5
votes
1
answer
247
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 $\...
- 3,011
5
votes
1
answer
291
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 ...
- 4,717
5
votes
1
answer
982
views
sets without perfect subset in a non-separable completely metrizable space
Suppose $X$ is a completely metrizable (but not separable) space. Suppose $D$ is a Borel (actually $F_{\sigma}$) subset of $X$. Is there any logical relation between the following statements?
[1] $D$...
- 51
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
...
- 53
5
votes
1
answer
584
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_{\...
- 41.8k
5
votes
1
answer
528
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 $\mathcal ...
- 41.8k
5
votes
1
answer
479
views
A question about Q?
Let A=$\{a_n : n\in \omega \}\subset 2^{\omega\times\omega}$ be nonempty countable without isolated points (i.e. homeomorphic to $\mathbb{Q}$), and satisfy $ \forall n\in \omega \exists^\infty m|\{k:...
- 427
5
votes
1
answer
254
views
Is the topology of weak+Hausdorff convergence Polish?
Let $X$ be a compact metric space, $P_X$ the set of Borel probability measures on $X$, and $K_X$ the set of non-empty closed subsets of $X$. I will define the "topology of weak+Hausdorff ...
- 708
5
votes
3
answers
402
views
Measure on hyperspace of compact subsets
For a Polish space $X$, let $K(X)$ be the set of compact subsets of $X$. Given the topology with basis $\{K\in K(X):K\subset U_0, K\cap U_1\neq\emptyset,\ldots,K\cap U_n\neq\emptyset\}$ for open sets ...
- 1,052
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 ...
- 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 ...
- 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\...
- 21.2k
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 ...
- 41.8k
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 ...
- 41.8k