# Generalizing topological spaces

I was studying topological spaces having studied metric spaces. I guess the properties of open sets in metric space led into (in terms of open set) definition of Topological spaces. Although there are many equivalent definitions such as Kuratowski closure axioms, the most used definition is in terms of open sets.

I know that we can extend these topological spaces into vector spaces or any other algebraic structures by defining appropriate operations in the given topological space. I'm not aware of any generalization of topological spaces which further broadens the definition of topological spaces.

For example, if we replace union and intersection by any other two operations (let's call it op1 and op2) such that arbitrary op1 and finite op2 are closed in the given class of subsets of a set and conventionally we can call them as open$$^*$$ sets(I'm calling them open$$^*$$ to distinguish from the open sets in the normal sense). This idea is itself is worth reflecting because if we consider op1 as intersection and op2 as union then closed sets(in the normal sense) are open$$^*$$ sets and open sets(in the normal sense) are closed$$^*$$ sets. If we consider some other operations then the properties will differ entirely. In general, it may not lead us into metric spaces or may not have pleasant properties which always is the aim of mathematicians.

Does this kind of generalization of topological space exist? Did anyone work in this direction?

• Do you know Stone duality? It deals with something similar, namely finding algebraic structures which behave as if they were the sets of open/closed sets of a topological space. Nov 23 '20 at 13:16
• I don't know about "revising" topological notions by using set operations to replace union and intersection, but if one were to venture down this road I'd recommend looking at Hausdorff operations (see also this search and this search). For the usual generalizations of topology, see my answers to this question, especially the references. Nov 23 '20 at 14:01
• Regarding generalizations and extensions of Hausdorff operations, see Kolmogorov's ideas in the theory of operations on sets by Vladimir Grigorevich Kanovei (1988) and Memoir on the analytical operations and projective sets (I) (II) by Kantorovitch/Livenson (1932, 1933). Nov 23 '20 at 14:12