This property seems like it should have a nice name, but I can't find one anywhere. Does anyone know a name for this?

For each non-empty open set $U$, there exist proper open subsets $\{U_i\}_{i\in I}$ such that $U=\cup_i U_i$.

I suppose this could also be formulated as each nonempty open set having an open cover of proper subsets, or being the colimit of its open subsets.

(Also, apologies if this is something obvious I should have thought of.)

nonemptyopen set? – user2734 Mar 19 '10 at 16:41caringwhat the name of such a space is. I mean, if you have examples of some of these spaces, and some result that says that this precise property is what you need for some application, then by all means, it should have a name, and knowing the conventional name will help you look up the appropriate literature. But as it is, I'd like some motivation before I'll like the question. – Theo Johnson-Freyd Mar 20 '10 at 1:43