Suppose we have $\cup, \cap$ defined as binary operations on "subsets" of some set $S$, where "subsets" can be something exotic like substrings of a string. Suppose also that they satisfy analogous axioms for defining a topology on $S$.
Then I think the structure could be called a semiring as $S$ becomes the multiplicative ($\cap$) identity and $\varnothing$ becomes the additive ($\cup$) identity. Suppose also that there exist a set difference operator that acts just like that between sets.
What can we say about the "topological side" of things if even possible. Thanks.