An α-disconnected space has no proper monic preimage

Abstract

All spaces are compact Hausdorff. α is an uncountable cardinal or the symbol ∞. A continuous map τ: X→ Y is called an α-SpFi morphism if τ -1( G) is dense in X whenever G is a dense α-cozero set of Y. We thus have a category α-SpFi (spaces with the α-filter) which, like any category, has its monomorphisms; these need not be one-to-one. For general α, we cannot say what the α-SpFi monics are, but we show, and R.G. Woods showed, that ∞-SpFi monic means range-irreducible. The main theorem here is: X has no proper α-SpFi monic preimage if and only if X is α-disconnected. This generalizes (by putting in α = ∞) the well-known fact: X has no proper irreducible preimage if and only if X is extremally disconnected. If, in our theorem, we restrict to Boolean spaces and apply Stone duality, we have the theorem of R. Lagrange, that in Boolean α-algebras, epimorphisms are surjective. The theory of spaces with filters has a lot of connections with ordered algebra—Boolean algebras of course, but also lattice-ordered groups and frames. This paper is a contribution to the development of this topological theory.