Is there an abstract structure that characterizes connectedness, analogously to how topological spaces characterize continuity?


Here's one way to make this question more precise: if $(X,T\_X)$ is a topological space with underlying set $X$ and topology $T\_X$, then consider the pair $(X,C\_X)$ where $C\_X$ is the set of connected subsets of $X$. A continuous map $f:(X,T\_X) \to (Y,T\_Y)$ has the property that the direct image of a connected set is connected. Thus, we have a functor from **Top** to the category whose objects are pairs $(X,C\_X)$ of a set equipped with a set of subsets, and whose maps $f:(X,C\_X)\to (Y,C\_Y)$ are functions $f:X\to Y$ such that $f(U) \in C\_Y$ for all $U\in C\_X$. Is there a naturally defined full subcategory of the latter category that is "close" to the image of **Top**?


There are undoubtedly other ways that one could make this question more precise; if you have a better suggestion feel free to make it.



Abstract connectedness
There seems to be some literature on this already: this [paper](http://www.math.shimane-u.ac.jp/memoir/39/D.Buhagiar.pdf) introduces the notion of a "connective space", i.e., a set equipped with a "connectology", and develops some theory. There was some related Mathematics Stackexchange discussion [here](https://math.stackexchange.com/questions/187145/strengthening-of-a-kuratowskis-theorem-about-connected-sets?lq=1) which connects up with the nontrivial property mentioned by Eric Wofsey in a comment below the question. 


