Category Archives: Topology

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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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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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

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A complete, compatible metric for the strong topology (p15 (A))

Let be separable Banach spaces over . Show that the following is a complete, compatible metric for the strong topology on : where is a dense sequence in the unit ball of . Proof. It is not difficult to check … Continue reading

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Any subspace of a separable metric space is separable

Let be a separable, metric space, , and let be countable and dense.  For each , choose  if such intersection is nonempty.  Then,  is a countable subset of . Let . Then choose .  Now, choose such that .  Then, , so … Continue reading

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