Somewhat in line with this previous MathOverflow question:
I'm looking at a combinatorial structure consisting of a finite set $S$ of objects, and a family $F$ of designated subsets of $S$. We call elements of $F$ "connected" subsets. (Think of them like connected sets in a topological space, in a loose way.) This family has to satisfy two properties:
If any two connected sets have non-empty intersection, their union will be connected. (I call this property "glueability".)
Singletons are always connected. (This property is too vacuous to deserve a name.)
My question is whether any of you have seen anything like this before, or have any idea how to deal with it. This is a sort of awkwardly vague request to make, but I would be very interested to hear if this structure had arisen in any other contexts.
(As an example of what I've come up with: You can easily prove that the maximal connected sets exactly partition the whole set.)