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Results tagged with descriptive-set-theory
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Descriptive Set Theory is the study of definable subsets of Polish spaces, where definable is taken to mean from the Borel or projective hierarchies. Other topics include infinite games and determinacy, definable equivalence relations and Borel reductions between them, Polish groups, and effective descriptive set theory.
29
votes
Accepted
Closed balls vs closure of open balls
The following theorem (or its corollary) implies negative answer to the original question.
Theorem. For any point $x$ of a metric space $(X,d)$ the set $R_x:=\{r>0:cl(B(x,r))\ne \bar B(x,r)\}$ has …
- 42k
12
votes
Accepted
Is a Borel image of a Polish space analytic?
I have found a simple counterexample to Problems 1 and 2 (maybe it will be helpful to other researchers):
Fact. The Sorgenfrey line $\mathbb S$ does not have countable network (and hence is not a …
- 42k
11
votes
Accepted
Does a completely metrizable space admit a compatible metric where all intersections of nest...
Let us say that a topological space $X$ is spherically completely metrizable if the topology of $X$ is generated by a spherically complete metric.
Theorem. Every closed subspace $X$ of the countable …
- 42k
7
votes
Accepted
Meager subgroups of compact groups
This problem was been answered in negative by M.Laczkovich (http://www.ams.org/journals/proc/1998-126-06/S0002-9939-98-04241-5/S0002-9939-98-04241-5.pdf). He constructed a proper Borel subgroup $H$ of …
- 42k
6
votes
Accepted
On continuous perturbations of functions of the first Baire class on the Cantor set
After some thinking I realized that the answer to this question is negative. A counterexample can be constructed by a standard diagonal method of killing all possible candidatures.
We shall construct …
- 42k
6
votes
Accepted
Is every Borel function a projection of a Borel function with closed graph?
Yes, this is true: by Exercise 13.5 in Kechris' "Classical Descriptive Set Theory", for any Borel function $f:X\to Y$ between Polish spaces there exists a continuous bijective map $i:Z\to X$ from a Po …
- 42k
5
votes
Accepted
Are σ-sets preserved by Borel isomorphisms?
Under some set-theoretic assumptions the answer to this question is negative.
Namely, if there exists a $Q$-set $X$, then $X$ is a $\sigma$-set which is Borel isomorphic to a (hereditarily normal co …
- 42k
5
votes
Accepted
Is the Hilbert cube the countable union of punctiform spaces?
The Hilbert cube can be written as the union of two punctiform spaces. Just take any Bernstein set $X\subset[0,1]^\omega$ and observe that compact subsets in $X$ and $Y=[0,1]^\omega\setminus X$ are at …
- 42k
5
votes
Accepted
Topologically Ordered Families of Disjoint Cantor Sets in $I$?
Such an enumeration does not exist. To derive a contradiction, take any family of Cantor sets $(C_\alpha)_{\alpha\in A}$ in $[0,1]$, indexed by an uncountable subset $A\subset[0,1]$.
Let $\mathcal …
- 42k
5
votes
Accepted
Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{...
Yes, there exists such a function: Consider the real line as a linear space over the field $\mathbb Q$ and find a linearly independent Cantor set $C\subseteq \mathbb R$
(using the Kuratowski-Mycielski …
- 42k
5
votes
Accepted
Hausdorff quasi-Polish spaces
I hope that the following space $P\mathbb Q^\omega$ is second-countable and quasi-Polish but not Polish.
Let $\mathbb Q$ be the field of rational numbners endowed with the discrete topology. Then its …
- 42k
4
votes
Accepted
A reference to a theorem on the equivalence of ideals of measure zero in the Cantor cube
Finally I have found a good reference to this theorem, which can be easily derived from the following Theorem 2.12 in this paper of Akin:
Theorem (Akin, 1999). For any strictly positive continuous me …
- 42k
4
votes
Accepted
Equivalent of Lusin's Theorem in Borel setting
The answer is No.
A suitable counterexample can be constructed as follows.
On the real line $\mathbb R$ consider the equivalence relation $E=\{(x,y)\in\mathbb R\times \mathbb R:x-y\in\mathbb Q\}$. …
- 42k
4
votes
Accepted
Is the sumset of two Haar positive closed subsets of a Polish group non-meager?
I have just realized that this my question has a simple negative answer: Denote by $\mathbb R_+=[0,\infty)$ the half-line. Observe that the countable product of lines $G=\mathbb R^\omega$ is an Abelia …
- 42k
3
votes
Accepted
Hurewicz versus Wadge hierarchy of zero-dimensional Borel sets?
After some thoughts I realized that the answers to Problems 1 and 3 are negative. Namely, each limit Wadge class contains infinitely many Hurewicz non-equivalent spaces. Indeed, take a sequence $(U_n) …
- 42k