# 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

## Cantor-Bendixson 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 Cantor-Bendixson Theorem. Exercise  (6.6): is the largest perfect subset of . Proof. Suppose is … Continue reading

## 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

## 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

## 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