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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.
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The (global) theory of Borel equivalence relations
What do we know about the complexity of the theory of Borel equivalence relations, with the Borel reducibility order $\leq_B$?
That is, let $\mathcal{B}$ be the set of all Borel equivalence relation …
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Is there a standard Borel space of finitely branching real trees?
Given a set $X$, by a tree in $X$ I mean a set $T$ of finite sequences of elements of $X$ which is closed under initial segments. It is pruned of every element has a proper extension, and finitely bra …
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Is the set of subsequences of branches through a tree Borel?
Let $T$ be pruned subtree of $\omega^{<\omega}$. For my cases of interest, we may assume that $T$ is infinitely branching at every node, and consists of increasing sequences.
Let $A=\{x\in\omega^{\om …
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Set of perfect subsets of a Borel set
Let $\mathbb{P}$ be the set of all perfect (i.e., every node has incomparable successors) subtrees of the full binary tree $2^{<\omega}$. We can endow $\mathbb{P}$ with a Borel structure by considerin …
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Is the ideal of compact operators strongly Borel?
Let $H$ be a separable infinite dimensional Hilbert space. Denote by $\mathcal{B}(H)$ the space of bounded operators on $H$, and $\mathcal{K}(H)$ the ideal of compact operators. When endowed with the …
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A Borel perfectly everywhere surjective function on the Cantor set
Does there exist a Borel (or even continuous) function $f:\mathcal{C}\to\mathcal{C}$, where $\mathcal{C}$ is the Cantor set (or Cantor space $2^\omega$) such that for every nonempty closed perfect set …
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A Borel perfectly everywhere dominating family of functions
Is there a Borel function $f:2^\omega\to\omega^\omega$ such that for every nonempty closed perfect set $P\subseteq 2^\omega$, $f|P$ is a dominating family of functions in $\omega^\omega$?
This is a re …
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Competing definitions of smooth orbit equivalence relation
Suppose that $X$ is a standard Borel space (meaning it is endowed with a $\sigma$-algebra coming from some Polish topology on $X$) and $G$ is a Polish group acting in a Borel way on $X$. Denote by $E_ …
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Is the set of clopen subsets Borel in the Effros Borel space?
Let $X$ be a Polish space and $\mathcal{F}(X)$ the set of closed subsets of $X$ endowed with the Effros Borel structure, generated by sets of the form $\{F\in \mathcal{F}(X):F\cap U\neq \emptyset\}$, …
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Complexity of the set of closed subsets of an analytic set
Let $X$ be a compact Polish space and $K(X)$ the hyperspace of closed subspaces of $X$ with the Vietoris/Hausdorff metric topology.
Question: If $A$ is an analytic subset of $X$, what is the complexi …
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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 …
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Countable $\mathbf\Sigma^1_2$ equivalence relations
This question is meant to be viewed under moderate large cardinal hypotheses, e.g., enough to ensure $\aleph_1^{L[x]}<\aleph_1$ for all reals $x$.
In analogy with the (well-developed) theory of count …
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Convergence and winning strategies
Suppose we have a set $B\subseteq 2^\omega\times\omega^\omega$ and a sequence $(x_n)$ in $2^\omega$ such that for each $n$, Player I (the one trying to get into the payoff set) has a winning strategy …
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Products and Gale-Stewart games
For the purpose of this post, I will say that the Gale-Stewart game is the infinite two-player game of perfect information where players I and II alternate playing natural numbers, with I going first. …
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Countable companions for Polish locally compact groups and their orbit equivalence relations
In "Countable sections for locally compact group actions" (Ergod. Th. & Dynam. Sys., 1992), Kechris proved that if $G$ is a Polish locally compact group acting in a Borel way on a standard Borel space …
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