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Results tagged with descriptive-set-theory
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user 61536
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.
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
2
votes
Accepted
Can each non-open analytic subgroup of a Polish abelian group be covered by countably many c...
The answer to both problems (1 and 2) is negative: the Polish group $G=\mathbb Z^\omega$ contains a dense meager Borel subgroup $H$ (which can be written as the difference $H=A\setminus B$ of two $F_\ …
- 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
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
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
2
votes
Accepted
Bernstein sets of large cardinality
After thinking a night on this question and waking up, I realized that the answer is almost trivial: there are restrictions on possible cardinalities of generalized Bernstein set.
Any metrizable sp …
- 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
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
0
votes
A non-Borel union of unit half-open squares
Both problems have negative answer under CH and positive answer under $\neg$CH. The proofs can be found here.
- 42k
1
vote
Do continuous maps factor through continuous surjections via Borel maps?
Just a small addition to the existing answers.
Theorem. There exists a non-metrizable compact Hausdorff space $K$ admitting a continuous surjective function $f:K\to[0,1]^2$ to the unit square such …
- 42k
3
votes
Accepted
Is a locally finite union of $G_\delta$-sets a $G_\delta$-set?
I found a simple counterexample, which is however not regular.
Example. There exists a functionally Hausdorff second-countable space $X$ containing a closed discrete subset $D$, which is not of type …
- 42k
1
vote
Which topological spaces have a standard Borel $\sigma$-algebra?
Since every Polish space is Borel isomorphic to a zero-dimensional compact metric space, it suffices to characterize topological spaces, which are Borel isomorphic to a zero-dimensional compact metri …
- 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
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
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