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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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Good source for Effective Descriptive Set Theory
I just finished a first course in Descriptive Set Theory using Kechris' "Classical Descriptive Set Theory" and was hoping to find a good source for learning some of the Effective DST. Kechris doesn't …
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Measure on hyperspace of compact subsets
For a Polish space $X$, let $K(X)$ be the set of compact subsets of $X$. Given the topology with basis $\{K\in K(X):K\subset U_0, K\cap U_1\neq\emptyset,\ldots,K\cap U_n\neq\emptyset\}$ for open sets …
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Natural examples of $\bf\Sigma^0_3$ equivalence relations
I have been reading about Borel equivalence relations and I have noticed that while $\bf\Sigma^0_3$ equivalence relations are mentioned, there is a conspicuous absence of natural examples (other than …
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Descriptive Complexity of Knot Equivalence
I was reading a little about knots (in a popular math book that wasn't very good) and the book put forth several knot invariants like the Alexander and Jones polynomials. But these are not complete in …
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Is the Turing equivalence relation the orbit equiv. relation of the action of a countable gr...
The Turing equivalence relation on $\cal P(\mathbb{N})$ is defined by $A\equiv_T B$ iff $A\leq_T B$ and $B\leq_T A$. This is a countable Borel equivalence relation on the polish space $\cal P(\mathbb …
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Image of L^1 under the Fourier Transform
The Fourier Transform $\mathcal{F}:L^1(\mathbb{R})\to C_0(\mathbb{R})$ is an injective, bounded linear map that isn't onto. It is known (if I remember correctly) that the range isn't closed, but is d …
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