Extremally disconnected spaces as {{u->a,b<-v}-->{u->a=b<-v}}^l, and being proper as ({{o}-->{o->c}}^r_<4)^lr

Abstract

We observe that the notions of a topological space being extremally disconnected, and of a continuous map of compact Hausdorff spaces being proper, and being surjective proper, can each be defined in terms of the Quillen lifting property with respect to a surjective proper morphism of finite topological spaces, i.e. in terms of a monotone map of finite preorders. This reveals the preorders implicit in the statement of the Gleason theorem that extremally disconnected spaces are projective in the category of compact Hausdorff topological spaces, and interprets it as an instance of a weak factorisation system generated by an explicitly given morphism.