"Connection Space" 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?
 A: The answer is yes. See this article (J. Muscat & D. Buhagiar - Connective Spaces).
Added: I did some googling and found also this preprint (S. Dugowson - On Connectivity Spaces) which examines related ideas. The references here suggest that such spaces have been studied already by R. Börger in 1983.
A: In my book "Topology and Groupoids" I give one definition of a connected space as follows. Let $\mathbf 2$ be the discrete topological space on the set $\{0,1\}$. Then a space $X$ is connected in the usual sense if and only if every continuous function $X \to \mathbf 2$ is constant. This definition can be used for some neat proofs: see my answer to question 90746. Part of the point is to shift emphasis from the internal properties of the space to the view of the space in the category $\mathsf{ Top}$ of spaces and continuous functions. 
Notice also that a singleton space $S$ is a terminal object in $\mathsf{ Top}$ and $\mathbf 2= S \sqcup S$ (disjoint union) so one is near to a categorical definition of connectivity. 
