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Bases of a topological space in point set topology will in general form a coverage on its category of inclusion on open subsets and on its category of inclusion on basic opens, but it takes a bit more work to check whether either forms a Grothendieck pretopology. Is there a useful or natural criterion for when a (point-set) basis does give a (Grothendieck) basis?

The criteria may apply either to the bases themselves, or to any particularly nice property of a topological space that forces some class of bases to have that property.

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    $\begingroup$ Do you mean pretopology on the category of all open sets of the space or on its subcategory formed by the basic opens? (See my comment to the answer below) $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Feb 21, 2022 at 19:43
  • $\begingroup$ Good observation. As the question is written, it considers the category of all open sets of the space, but both cases are interesting. $\endgroup$
    – saolof
    Commented Feb 21, 2022 at 20:50
  • $\begingroup$ I edited the question to mention both cases. $\endgroup$
    – saolof
    Commented Feb 21, 2022 at 21:14

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This is a matter of expanding the definition, in this case Definition II.1.3 in SGA 4, which defines pretopologies.

By a “base” in this answer I mean what appears to be the most common definition: a collection of subsets of a fixed set $A$ such that any finite intersection of elements in the base is a union of elements in the base.

There are also multiple constructions of a site (i.e., a category with a coverage) from a base of a topological space $A$. One can either (A) take the category of all open subsets of $A$, or (B) the category whose objects are open subsets of $A$ that belong to the base. For covering families of some object $V$, one can either take (a) those open covers of $V$ whose elements belong to the base, or (b) open covers of $V$ whose elements are given by the intersection of $V$ and some element of the base. Altogether, there are three different options: A-a, A-b, B-a, and only option A-b produces a pretopology in the sense of SGA 4.

Axiom PT0 for pretopologies says that any morphism in a covering family admits base changes. Such base changes are always given by the corresponding intersection, provided that the intersection belongs to the category. Thus, PT0 is satisfied for options A-a, A-b and not satisfied for option B-a. If the base is closed under intersections, then PT0 is also satisfied for option B-a.

Axiom PT1 says that base changes of covering families are covering families. In our case, the base change is given by the intersection with some open subset $V$. Thus, PT1 is satisfied for option A-b and not satisfied for options A-a, B-a. If the base is closed under intersections, then PT1 is also satisfied for options A-a, B-a.

Axiom PT2 says that covering families can be composed. This is trivially true for the case under consideration.

Axiom PT3 says that the singleton family consisting of the identity map is a covering family, which is tautologically true in our case.

Thus, option A-b always gives a pretopology, whereas options A-a, B-a give a pretopology if and only if the base is closed under intersections of pairs.

In particular, open balls in a metric space as a coverage on the category of open subsets trivially form a pretopology using option A-b, and do not for a pretopology in options A-a, B-a.

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  • $\begingroup$ So would it also be correct to say that a base forms a pretopology iff it consists of the finite intersections of a subbase? $\endgroup$
    – saolof
    Commented Feb 21, 2022 at 19:27
  • $\begingroup$ If I understand the question correctly, it asks about pretopology on the category of all open sets rather than on its subcategory formed by basic opens. In that case, if one defines covers of $U$ as families $(B_i)_{i\in I}$ of basic opens $B_i\subseteq U$ whose union is $U$, I believe PT0 is also trivially satisfied. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Feb 21, 2022 at 19:33
  • $\begingroup$ @მამუკაჯიბლაძე: I added the other interpretation, just in case. $\endgroup$
    – Dmitri Pavlov
    Commented Feb 21, 2022 at 20:47
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    $\begingroup$ @saolof: The nLab claim is a mistake. There is no reason why the category of morphisms that belong to some covering family should have pullbacks, which is what this article appears to be asserting. $\endgroup$
    – Dmitri Pavlov
    Commented Feb 21, 2022 at 21:02
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    $\begingroup$ @მამუკაჯიბლაძე: Yes, so there are at least three different ways to produce a site from a base for a topological space. I adjusted my answer to treat all three variants. $\endgroup$
    – Dmitri Pavlov
    Commented Feb 21, 2022 at 22:07

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