I am interested in various generalizations of the notion of topological space; also in topologies placed in untypical frameworks, i.e. intuitionistic topological spaces or nano topological spaces. Recently, I've noticed that some authors started to investigate "binary" and "n-ary" topological spaces. The main idea is: while in the product topology we are working with the power-set of cartesian product (of spaces), then in binary / n-ary case we are interested in the cartesian product of power-sets. My question is quite a general one: do you think that this whole idea can be reliable or sensible?
Explanation: unfortunately, it seems that there is a lot of papers about "int. topo. spaces", "nano topo. spaces", "soft-sets topologies", "generalized topologies" (i.e. families closed under arbitrary unions without any additional conditions), "weak structures" etc., the same about binary topologies, but they are barely scratching the surface of the subject: namely, the authors often limit themselves to the presentation of "generalized" (or "intuitionistic", or "nano", or "binary") analogues of classical notions (various kinds of "continuity" and "convergence"). These papers are ok and I like them... but this whole branch of maths seems to be still on the initial stage. Examples and counter-examples are often based on simple and finite sets etc. Hence, have you met with more complex applications of these ideas? E.g. in formal logic, analysis, measure theory or in geometric / algebraic topology?
