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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) …
- 41.8k
4
votes
1
answer
129
views
Hurewicz versus Wadge hierarchy of zero-dimensional Borel sets?
Given two subsets $A,B$ of the Cantor cube $2^\omega$ we write $A\le_W B$ (resp. $A\le_H B$) if there is a continuous (and injective) function $f:2^\omega\to 2^\omega$ such that $f^{-1}(B)=A$. The rel …
- 41.8k
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_\ …
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11
votes
0
answers
142
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 conditi …
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6
votes
1
answer
242
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).
I …
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3
votes
0
answers
72
views
Borel complexity of special unions of Polish spaces
Let $X$ be a compact metrizable space and $(A_q)_{q\in\mathbb Q}$ be a family of pairwise disjoint sets, indexed by rational numbers. Assume that the family $(A_q)_{q\in\mathbb Q}$ has the following p …
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7
votes
1
answer
283
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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
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 …
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2
votes
0
answers
96
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Codimension of analytic linear subspaces in Polish vector spaces
Let $A$ be a linear analytic subspace of a Polish vector space $X$.
Using Piccard-Pettis Theorem, it is easy to prove that $A$ has uncountable codimension in $X$. If $A$ is of type $F_\sigma$, then ap …
- 41.8k
6
votes
1
answer
172
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Can each non-open analytic subgroup of a Polish abelian group be covered by countably many c...
By a result of Laczkovich ('Analytic subgroups of the reals' Proc AMS Vol 126 (1998)), any non-open analytic subgroup of a Polish locally compact group can be covered by countably many closed Haar nul …
- 41.8k
7
votes
0
answers
168
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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 …
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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 …
- 41.8k
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 …
- 41.8k
6
votes
1
answer
328
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 a ( …
- 41.8k
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 …
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