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.

Axiom PT0 for pretopologies says that any morphism in a covering family admits base changes.
This means that for any inclusions $U→X$, $V→X$ of opens in a base, the intersection $U\cap V$ also belongs to the base.

Axiom PT1 says that pullbacks of covering families are covering families.
This is always true because of the previous axiom and the fact that unions commute with intersection with a fixed open subset.

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, a base for a topological space is a pretopology if and only if it is closed under intersection of pairs.

In particular, open balls in a metric space in general do not form a pretopology, since the intersection of two open balls is not an open ball in general.