What are the names for the structures obtained when we drop some topological space axioms? 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?
 A: The notion of a forgetful functor is the right mathematical setting for your question.
Rephrasing your examples, there is a natural forgetful functor from the category of groups to the category of monoids, another one from the category of monoids to the category of semigroups and yet another one from the category of semigroups to the category of magmas.
However, forgetful functors do not really answer your question, since they do not infer in general any new terminology. (I added the category-theory tag to your question to see whether category theorists have further insight on your question).
Coming back to your last example, I don't know if there is a specific name for the structure you describe. However, there is a name for a slightly more restricted structure: a Pervin space is a pair $(X, \mathcal{L})$ where $X$ is a set and 
$\mathcal{L}$ is a lattice of subsets of $X$, that is, a subset of $\mathcal{P}(X)$ closed under finite unions and finite intersections. Contrary to topological spaces, the elements of $\mathcal{L}$ are not supposed to be closed under arbitrary unions, but just under finite ones. A morphism from the Pervin space $(X, \mathcal{L}_X)$ to the Pervin space $(Y, \mathcal{L}_Y)$ is a map $f: X \to Y$ such that, for every $S \in \mathcal{L}_Y$, $f^{-1}(S) \in \mathcal{L}_X$. 
