Examples of topological spaces with canonical bases with the following property (redivisibility)

Solution 1:

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.

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