Uniformities in the Descriptive Theory of Sets I: Basic Operators

Abstract

The purpose of this paper is to set forth connections which the author has discovered between the descriptive theory of sets and concepts currently under consideration in the theory of uniform spaces. We comment that the first connections of this nature have been obtained by Z. Frolik in [Fr]1 and by A. W. Hager in [H]3. The results of this paper and a companion paper further confirm that each subject sheds light on problems in the other area. For example, in uniform theory it has previously seemed difficult to construct measurable spaces that were not locally fine; Theorem 2.3 shows that one may use the measurable space associated with a complete metric space in which there exists bi-analytic, non-Baire sets. Other examples are provided by the following results, which will appear in the companion paper ([R]4): (i) The metric-valued Baire measurable functions on a complete metric space are precisely the functions that are uniformly continuous with respect to the associated measurable spaces, and (ii) if the measurable space associated with a uniform space is proximally fine, then the graph of each metric-valued Baire mapping is a Baire set. The following is a brief sketch of the contents of the present paper. The first result, Theorem 2.1, shows that the measurable space associated with a uniform space coincides with the uniform space derived from the disjoint completely additive Baire covers precisely when the measurable space is finest in its proximity class. Theorem 2.2 shows that for each complete metric space the associated measurable space is finest in its proximity class precisely when the Baire v-field is proximally fine. Theorem 2.3 shows that Lusin's First Separation Theorem is valid in a complete metric space exactly when the associated measurable space is locally fine, which occurs precisely when the Baire v-field is h-closed. This result should be the starting point for an investigation (not yet initiated) which seeks to equate degrees of the locally fine property with partial