LIPSCHITZ FUNCTIONS ON TOPOMETRIC SPACES

Description

LIPSCHITZ FUNCTIONS ON TOPOMETRIC SPACES ITAÏ BEN YAACOV Abstract. We study functions on topometric spaces which are both (metrically) Lipschitz and (topo- logically) continuous, using them in contexts where, in classical topology, ordinary continuous functions are used. (i) We define normal topometric spaces and characterise them by analogues of Urysohn's Lemma and Tietze's Extension Theorem. (ii) We define completely regular topometric spaces and characterise them by the existence of a topo- metric Stone-?ech compactification. (iii) For a compact topological space X, we characterise the subsets of C(X) which can arise as the set of continuous 1-Lipschitz functions with respect to a topometric structure on X. Introduction Topometric spaces are spaces equipped both with a metric and a topology, which need not agree. To be precise, Definition 0.1. A topometric space is a triplet (X,T , d), where T is a topology and d a metric on X, satisfying: (i) The distance function d : X2 ? [0,∞] is lower semi-continuous in the topology. (ii) The metric refines the topology. Compact topometric spaces were first defined in [BU10] as a formalism for various global and local type spaces arising in the context of continuous first order logic, allowing for some kind of (topometric) Cantor- Bendixson analysis in spaces which, from a purely topological point of view, are possibly even perfect.

• then


• normal topometric


• closed metric


• metric neighbourhoods


• completely regular


• every normal


• all continuous


• space


• all c?

Sujets

• Tychonoff space

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 15

Extrait

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