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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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Almost universal properties
Suppose that a small category $C$ has sets of objects and arrows that carry the structure of, say, Polish spaces, in some appropriately compatible way. For example $C$ might be an internal category in …
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Games that never begin
Games that never end play a major role in descriptive set theory. See for example Kechris' GTM.
Question: Does there exist a literature concerning games that never begin?
I have in mind two playe …
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Are Vitali-type nonmeasurable sets determinate?
Here, by a Vitali set, I mean the following. Call $f_1,f_2:\omega\rightarrow 2$ tail-equivalent if {$n| f_1(n)\not=f_2(n)$}$<\infty$. Vitali sets (existence via AC) contain one such $f$ from each …
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Closure properties of familes of $G_\delta$ sets.
Given a family of sets $G\subset P(X)$, can one characterize by "closure properties" alone whether or not $G$ arises as the family of all $G_\delta$ for some topology on $X$? some Polish space topolo …
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A strange planar set and the Continuum Hypothesis
Call a number abnormal if its decimal expansion doesn't feature every digit an infinite number of times. Call a triangle in ${\Bbb R}^2$ abnormal if at least one of its angles spans an abnormal frac …
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Is every $\sigma$-algebra of sets *abstractly* the Borel algebra of a topology on perhaps so...
Is every sigma-algebra the Borel algebra of a topology?
inspires the present question which asks for less.
Question: Given a $\sigma$-algebra $\mathcal A$ on a set $X$, does there exist a topology $\m …
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