Functional topology for geometric settings

In this paper we propose two approaches to connect the functional topology theory of Clementino et al. (2004) [4], with the situation in the category of schemes. In a first approach, we extend an example from [4], to a more general pattern for obtaining a class of “closed morphisms” from two auxiliary classes, “closed embeddings” and newly added “surjections”. In a second approach we apply [4] to the presheaf category over , and recover the notions of properness and separatedness on .