The topological shadow of \({{{\mathbb {F}}}_1}\)-geometry: congruence spaces

In this paper we introduce congruence spaces, which are topological spaces that are canonically attached to monoid schemes and that reflect closed topological properties. This leads to satisfactory topological characterizations of closed morphisms and closed immersions as well as separated and proper morphisms. We study congruence spaces thoroughly and extend standard results from usual scheme theory to monoid schemes: a closed immersion is the same as an affine morphism for which the pullback of sections is surjective; a morphism is separated if and only if the image of the diagonal is a closed subset of the congruence space; a valuative criterion for separated and proper morphisms.