Somewhat in line with this previous MathOverflow question:

I'm looking at a combinatorial structure consisting of a set of objects $S$, and a family of subsets of $S$ which are "connected". 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.)