Bases of a topological space in point set topology will in general form a coverage on its category of open subsets, not a Grothendieck pretopology. Is there a useful or natural criterion for when a (point-set) basis does give a (Grothendieck) basis for Grothendieck pretopology?

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.

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.

Bases of a topological space in point set topology will in general form a coverage on its category of open subsets, not a Grothendieck pretopology. Is there a useful or natural criterion for when a (point-set) basis does give a (Grothendieck) basis for Grothendieck pretopology?

The criteria may apply either to the bases themselves (i.e. anything that requires a basis to be a basis for the subspace topology of any open subset), or to any particularly nice property of a topological space that forces some class of bases to have that property.

Bases of a topological space in point set topology will in general form a coverage on its category of open subsets, not a Grothendieck pretopology. Is there a useful or natural criterion for when a (point-set) basis does give a (Grothendieck) basis for Grothendieck pretopology?

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.