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?


There are plenty of generalisations of topological spaces:

For example, NLab have pages on proximity spaces, pseudo-topological spaces, pretopologies and also Locales and Topoi. And the latter generalise to n-Topoi upto n=infinity.

There are also uniform spaces which generalise the notion of a space where uniform convergence applies; whilst Cauchy spaces generalise those spaces where completeness applies.

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    $\begingroup$ General Topology by R. Engelking. Out of 8 chapters, ch. 7 is Uniform Spaces & ch. 8 is Proximity Spaces. $\endgroup$ Aug 12 at 11:10

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