Relational structures associated to topological spaces and preserved by image functions

We study relational and algebraic first-order structures on $\mathcal P(X)$, for $X$ a topological space, with the further requirement that such structures are preserved by image functions associated to continuous functions. Many of the above structures have arisen independently in disparate and very distant fields. In particular, we deal with a ternary relation $ x \sqsubseteq^{\textstyle z} y$ whose intended interpretation is $x \subseteq z \cup Ky$, where $K$ is closure in some topological space. The study provides a smoother, simpler and more general theory, with respect to the formerly studied ``basic'' binary relation given by $x \subseteq Ky$. We provide an axiomatization for semilattices with such an ``extended'' ternary relation, characterizing those structures which can be embedded into a topological model with the above interpretation. More generally, we construct ``free extensions'' of extended specialization semilattices into closure semilattices. We also take into account the possibility of adding contact and $n$-ary hypercontact relations. In this way we generalize and uniformize many previous results.