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 sort of an awkwardly vague request to make, but I would be very interested to hear if this structure had arisen in any other contexts.