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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 aksing 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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# 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 aksing 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.