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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.
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
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
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
2
votes
Accepted
Is the topology of weak+Hausdorff convergence Polish?
The set $MS=\{(\mu,K)\in P_X\times K_X:\mathrm{supp}(\mu)=K\}$ is of type $G_\delta$ in $P_X\times K_X$ and hence the weak+Hausdorff topology on $P_X$ is Polish.
Indeed, fix any countable base $\{U_n\ …
- 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
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
3
votes
Accepted
The Borel class of a countable union of $G_\delta$-sets, which are absolute $F_{\sigma\delta}$
The answer to this question is negative and follows from
Theorem. Each $G_{\delta\sigma}$-subset $A$ of a Polish space $X$ can be written as the union $\bigcup_{n\in\omega}A_n$ of a sequence $(A_n)_{n …
- 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
1
vote
Accepted
Is there a universally meager air space?
Lyubomyr Zdomskyy (in private communication) sent me the proof of the following result giving a consistent answer to Problems 1 and 2.
Theorem (Zdomskyy). Under $\mathfrak b=\mathfrak c$ there exists …
- 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
1
vote
Accepted
K-analytic spaces whose any compact subset is countable
I looked at the paper of Fremlin and have seen that a minor modification of his example yields the following theorem showing that my question is independent of ZFC.
Theorem. The following statemen …
- 42k
2
votes
Small uncountable cardinals related to $\sigma$-continuity
It seems that the equality $\sigma=\aleph_1$ established by Will Brian in the Cohen model, can be derived from the following upper bound for $\sigma$, which answers Problem 4.
Theorem 1. $\sigma\l …
- 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
3
votes
If non-empty player has a winning strategy in Banach-Mazur game BM(X), then it also has in B...
Topological spaces $X$ for which the second player (Non-empty) has a winning strategy in the Banach-Mazur game $BM(X)$ are called weakly $\alpha$-favorable by White and
Choquet by Kechris.
Accordin …
- 42k