The Stone–Čech Compactification and the Cozero Lattice in Pointfree Topology

Abstract

The existence of the pointfree Stone–Cech compactification, here understood as the compact, completely regular coreflection of a frame, has been established long ago in several ways modelled after certain descriptions of the classical Stone–Cech compactification (Johnstone, Stone Spaces, Cambridge Stud. Adv. Math. 3. Cambridge University Press, Cambridge, 1982; Banaschewski and Mulvey, Houston J. Math. 6:301–312, 1980, J. Pure Appl. Algebra 33:107–122, 1984), but – somewhat surprisingly – one of the most familiar representations of the latter, the one popularized by Gillman and Jerison’s influential book as the space of maximal z-filters, has only recently been recognized also to carry over to the pointfree setting, as is implicit in Wei (Appl. Categ. Structures 12:197–202, 2004). The purpose of this note is to present an alternative account of this fact which places the problem into the wider context of the saturation quotients of compact frames and what are introduced here as completely normal lattices. Specifically, we first show that the saturation quotient S(Open image in new window A) of the frame Open image in new window A of ideals of a completely normal lattice A is compact completely regular (Proposition 1) which is then brought to bear on the question at hand by establishing that the cozero lattice CozL of a frame L is completely normal and finally leads to the result that SOpen image in new window(CozL) is the Stone–Cech compactification of L (Proposition 2). Given that familiar results concerning saturation quotients exhibit the spectrum of the latter frame as the space of maximal z-filters of X for the cozero set lattice of a topological space X, this is then exactly the pointfree form of the desired kind. With regard to foundations, it should be emphasized that, in notable contrast with most other work involving CozL, the present arguments are constructively valid in the sense of topos theory.