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 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 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.)

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.)