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,
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.