Let us say that a family of sets $\mathcal{S}$ is redivisible if, for any $S_1, S_2\in \mathcal{S}$, the intersection $S_1\cap S_2$ is the union of finitely many sets from $\mathcal{S}$.
Which (classes of) topological spaces have a canonical basis which is redivisible?
Some examples I came up with:
- Order topology, with basis given by all intervals.
- Preorder topology, with basis given by upper sets.
- Ultrametric spaces, with basis given by open balls.
- While metric spaces don't seem to have this property in general, the euclidean topology with basis given by all open boxes does have it.
Is there any other notable example?
This is a vague question where by canonical I mean the following: definitional basis, a basis that you would give when describing/defining the topology.
As pointed out in the comments by Antonio this question is void/meaningless. If $(X,\tau)$ is any space, then $\mathcal{S}=\tau$ is a basis (you can call it canonical, which is itself a vague undefined term anyway) and it's trivially redivisible as it's closed under finite intersections, being a topology.