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Category Archives: Descriptive Set Theory
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
Posted in Analysis, Definition, Descriptive Set Theory, Kechris, Topology
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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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Questions
I have been unable to solve the following problems from Kechris. Any hints, advice, or solutions would be appreciated. 1. Exercise 4.9: Let be separable Banach spaces. The weak topology on is the one generated by the functions (from into … Continue reading
Posted in Analysis, Descriptive Set Theory, Kechris, Metric Spaces, Topology, Unsolved
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