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Results tagged with descriptive-set-theory
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user 12978
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.
2
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Classify spaces that make extension theorems hold
Recall a Polish space is a completely metrizable separable space.
Say a Polish space $Y$ is a terminal space if for any Polish space $X$ and any closed $C \subseteq X$, one can extend a continuous m …
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15
votes
2
answers
3k
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Generalizations of the Tietze extension theorem (and Lusin's theorem)
I am reasking a year-old math.stackexchange.com question asked by someone else.
(For my needs every space $X$ and $Y$ will be Polish---that is a completely separably metrizable space.)
The Tietze ex …
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Accepted
Is the following product-like space a Polish space?
Yes, $\mathbb X$ is a Polish space, even a computable one, but it doesn't look like it has a nice metric.
(I figured out the answer on my own question, but any other insightful answers or references …
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4
votes
Continuity on a measure one set versus measure one set of points of continuity
If $X$ and $Y$ are Polish and $Y$ is compact, then YES. (I think my proof can be fixed to handle the noncompact setting, but I don't see how right now.)
My proof involves this lemma.
Lemma. Assume …
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4
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1
answer
214
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Is the following product-like space a Polish space?
Let $\mathcal{M}_1(\mathbb R)$ denote the space of Borel probability measures on $\mathbb R$. The space is a Polish space (a space which admits a complete, separable, metric) using, say the Levy-Prok …
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2
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1
answer
952
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Is every path connected space continuously path connected
Recall a topological space $X$ is path connected if for all $x,y \in X$ there is a continuous function $f\colon [0,1] \to X$ such that $f(0)=x$ and $f(1)=y$.
Say that $X$ is continuously path connect …
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