Is there an established or well justified terminology for a topological base that is closed under finitary intersections?

As motivation, recall these conditions on a collection of subsets of a given set:

- closed under finitary intersections and arbitrary unions,
- closed under finitary intersections,
- filtered downwards,
- arbitrary.

Anything that satisfies one condition satisfies any later condition; conversely, anything that satisfies one condition generates something that satisfies any earlier condition. I know names for (1,3,4): ‘topology’, 'topological base' (or ‘base for a topology’), and ‘topological subbase’ (at least when thought of in this context). So I'm asking for a name for (2). And one reason that this is interesting is that the obvious way to generate (3) from (4) already gives you (2), so it really does come up.

couldalso introduce a distinction between base and basis here, although I couldn't recommend that. – Toby Bartels Oct 9 '11 at 6:02