Exact approximations to Stone–Čech compactification

Given a locale L and any (possibly empty) set-indexed family of continuous mappings F≡{fi}i∈I, fi:L→Li with compact and completely regular co-domain, a (generalized) compactification η:L→Lγ of L is constructed enjoying the following extension property: for every fi∈F a unique continuous mapping exists such that . Considered in ordinary set theory, this compactification also enjoys certain convenient weight limitations.Stone–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 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.A constructive (intuitionistic and predicative) proof that the class is a set whenever L is locally compact and L′ is set-presented and regular (in particular 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.