MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.