Sets completely separated by functions in Bishop Set Theory

TL;DR

This paper explores the structure of completely separated sets in Bishop Set Theory, introducing new concepts like global families and presenting set-theoretic analogs of classical topological theorems.

Contribution

It introduces the notion of global families of completely separated sets and provides set-theoretic versions of Stone-ch and Tychonoff theorems within Bishop Set Theory.

Findings

Defined global families of completely separated sets.

Presented set-theoretic versions of classical theorems.

Constructed the free completely separated set.

Abstract

Within Bishop Set Theory, a reconstruction of Bishop's theory of sets, we study the so-called completely separated sets, that is sets equipped with a positive notion of an inequality, induced by a given set of real-valued functions. We introduce the notion of a global family of completely separated sets over an index-completely separated set, and we describe its Sigma- and Pi-set. The free completely separated set on a given set is also presented. Purely set-theoretic versions of the classical Stone-\v{C}ech theorem and the Tychonoff embedding theorem for completely regular spaces are given, replacing topological spaces with function spaces and completely regular spaces with completely separated sets.