An equivalence relation for compact Hausdorff varieties

Abstract

The usual framework for Frechet equivalence of functions entails a Peano space, a metric space, and the class of continuous functions from the Peano space to the metric space [1]. We show that an equivalence relation can be defined in a more general context entailing functions from a compact Hausdorff space to a set, which need not have a topology, provided the functions satisfy a certain compatibility condition. The class of continuous functions from a compact Hausdorff space to a topological space is compatible, if in the latter space every compact set is closed. Therefore it is meaningful to consider compact Hausdorff varieties, which of course include Peano varieties, in topological spaces considerably more general than metric spaces. The formulation is of such a nature that the equivalence of two functions can be tested without reference to any range space topology. For separated uniform spaces equivalence may be reexpressed in terms of the uniformity. Within the Frechet framework this equivalence is identical to Frechet equivalence. As the basic structure we take a compact Hausdorff space X with the collection of closed sets 3, and a class of functions 9Y each of which is defined on X and takes values in a set Y, such that the following compatibility condition is satisfied: for all f, gEl and all ECX, if EEa then flg(E) Ea. For every function fEl: we define a topology af for its range in the following standard manner: for all E Cf(X), EE af if and only if f-1(E)E3.