On the existence of Stone-Cech compactification

TL;DR

This paper investigates the conditions under which the Stone-Cech compactification exists within constructive set theories and shows its existence is independent of certain axioms, contrasting with classical topos-theoretic results.

Contribution

It proves that the characterization of locales admitting Stone-Cech compactification holds without choice principles in CZF, and demonstrates the independence of its existence from axioms in constructive frameworks.

Findings

The class of continuous maps from compact regular locales to set-presented locales is a set in CZF.

Existence of Stone-Cech compactification for Boolean locales is independent of CZF+REA.

The characterization of locales with Stone-Cech compactification does not depend on choice principles.

Abstract

In [G. Curi, "Exact approximations to Stone-Cech compactification'', Ann. Pure Appl. Logic, 146, 2-3, 2007, pp. 103-123] a characterization is obtained of the locales of which the Stone-Cech compactification can be defined in constructive type theory CTT, and in the formal system CZF+uREA+DC, a natural extension of Aczel's system for constructive set theory CZF by a strengthening of the Regular Extension Axiom REA and the principle of dependent choice. In this paper I show that this characterization continues to hold over the standard system CZF plus REA, thus removing in particular any dependency from a choice principle. This will follow by a result of independent interest, namely the proof that the class of continuous mappings from a compact regular locale X to a regular a set-presented locale Y is a set in CZF, even without REA. It is then shown that the existence of Stone-Cech compactification of a non-degenerate Boolean locale is independent of the axioms of CZF (+REA), so that the obtained characterization characterizes a proper subcollection of the collection of all locales. The same also holds for several, even impredicative, extensions of CZF+REA, as well as for CTT. This is in contrast with what happens in the context of Higher-order Heyting arithmetic HHA - and thus in any topos-theoretic universe: by constructions of Johnstone, Banaschewski and Mulvey, within HHA Stone-Cech compactification can be defined for every locale.