Exact approximations to Stone–Čech compactification

Given a locale L and any (possibly empty) set-indexed family of continuous mappings F ≡ { f i } i ∈ I , f i : L → L i with compact and completely regular co-domain, a (generalized) compactification η : L → L γ of L is constructed enjoying the following extension property: for every f i ∈ F a unique continuous mapping f i γ : L γ → L i exists such that f i γ ∘ η = f i . Considered in ordinary set theory, this compactification also enjoys certain convenient weight limitations. Čech compactification is obtained as a particular case of this construction in those settings (such as ZF, or, more generally, topos theory) in which the class Hom Loc ( L , [ 0 , 1 ] ) of [0,1]-valued continuous mappings is a set for all L . This will follow by the proof that–also in the point-free context–a compactification that allows for the extension of [0,1]-valued mappings suffices for deriving the full reflection. tructive (intuitionistic and predicative) proof that the class Hom Loc ( L , L ′ ) is a set whenever L is locally compact and L ′ is set-presented and regular (in particular Hom Loc ( L , [ 0 , 1 ] ) for L locally compact) is also obtained; together with the described compactification, this makes it possible to characterize the class of locales for which Stone–Čech compactification can be defined constructively.