# Category Archives: Kechris

## 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

## 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