Motivation: If I start with the group axioms and drop the requirement that I have inverses, I get the monoid axioms. If I proceed to drop the requirement that I have an identity, I get the semigroup axioms. If I then drop the requirement of associativity, I get the magma axioms. If I drop the operation, I get the set axioms.
A map preserving the monoid structure is a "monoid homomorphism;" a map preserving the semigroup structure is a "semigroup homomorphism;" etc.
Question: Now suppose I start with the topological space axioms and start dropping conditions. Do the resulting sets of axioms have names? What about the maps preserving such structure -- do they have names? In particular, what about the smallest case of a sets equipped with some subset of their powersets, together with functions such that the preimage of a designated set is a designated set?