Can we distill the idea of "connectivity" away from their topological context and study abstract properties of "connectivity"?
I define a connective space to be a set $X$ together with a collection $\gamma$ of subsets of $X$, which we define as "connected". $\gamma$ contains every singleton subset of $X$, and for all $A, B \in \gamma$ such that $A \cap B \neq \emptyset$ we have $A \cup B \in \gamma$.
It might be interesting to study functions between connective spaces that preserve connected sets. Or, more suggestively, perhaps functions such that every pre-image of a connected set is connected... Does this exist in literature?