# Terminology for topological base closed under intersection?

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:

1. closed under finitary intersections and arbitrary unions,
2. closed under finitary intersections,
3. filtered downwards,
4. 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.

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It also comes up in that the collection of distinguished open subsets of an affine scheme satisfy (2), not just (3). I know that Kempf uses this fact in his paper "Some elementary proofs of basic theorems in the cohomology of quasi-coherent sheaves." Unfortunately, he uses the term "basis" to denote (2). –  Charles Staats Aug 16 '10 at 0:24
An (intersection) semilattice ? –  Gerald Edgar Oct 8 '11 at 19:26
A semilattice base? –  Toby Bartels Oct 9 '11 at 6:02
One could also introduce a distinction between base and basis here, although I couldn't recommend that. –  Toby Bartels Oct 9 '11 at 6:02