On the existence of Stone-Čech compactification

Abstract

Introduction. In 1937 E. Čech and M.H. Stone, independently, introduced the maximal compactification of a completely regular topological space, thereafter called Stone-Čech compactification [8, 23]. In the introduction of [8] the non-constructive character of this result is so described: “It must be emphasized that β(S) [the Stone-Čech compactification of S] may be defined only formally (not constructively) since it exists only in virtue of Zermelo's theorem”.By replacing topological spaces with locales, Banaschewski and Mulvey [4, 5, 6], and Johnstone [14] obtained choice-free intuitionistic proofs of Stone-Čech compactification. Although valid in any topos, these localic constructions rely—essentially, as is to be demonstrated—on highly impredicative principles, and thus cannot be considered as constructive in the sense of the main systems for constructive mathematics, such as Martin-Löf's constructive type theory and Aczel's constructive set theory.In [10] I characterized the locales of which the Stone-Čech 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.