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Author Archives: alanmath
Classification of smooth equivalence relations
The classification of smooth equivalence relations is accomplished by two exercises from Su Gao’s book, Invariant Descriptive Set Theory. Exercise 5.4.1, Su Gao, p 132. Show that for any equivalence relation on a Polish space , iff there are infinitely … Continue reading
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The Axiom of Choice and indeterminacy
It’s been a while since my last post — I apologize to any devoted fans who have been disappointed. I’m taking a class on Descriptive Set Theory now. Here is the proof of an interesting theorem from class: Assuming the … Continue reading
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CantorBendixson decompositions. Kechris, Exercise (6.6)
Kechris, p. 32, Exercise (6.6) (A) Let be a Polish space. Decompose uniquely into , where is perfect, is countable open. This decomposition is possible by CantorBendixson Theorem. Exercise (6.6): is the largest perfect subset of . Proof. Suppose is … Continue reading
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Hausdorff metric compatible with Vietoris topology / Kechris p25 B Ex 4.21
Kechris, p. 25 (B) Exercise (4.21). Show that the Hausdorff metric is compatible with the Vietoris topology. Proof. Let be a metric space with . Let denote the topology from the Hausdorff metric on , with basis , denote the … Continue reading
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Topological limits / Kechris p26, Ex 4.23
See here for definitions. Kechris, p. 26 (B) Exercise (4.23). Let be metric space with . Show that for nonempty , (i) . Proof. Suppose . Thus, for all , there exists , such that implies . That is, . … Continue reading
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Vietoris topology and Hausdorff metric
From Kechris: Let be a topological space. We denote by the space of all compact subsets of equipped with the Vietoris topology. Let be a metric space with . We define the Hausdorff metric on , , as follows: , … Continue reading
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Uniform metrics define same topology on C(X,Y)
Kechris, p. 24 (B) Let be a compact metrizable space and a metrizable space. We denote by the space of continuous functions from into with the topology induced by the uniform metric: where is a compatible metric for . A … Continue reading
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