Can we restrict the interpretation of ' belongs to' relation in set theory?

Question

In axiomatic set theory, the relation ' belongs to' is undefined, subject to the restriction imposed by other axioms, for example, the Axiom of Regularity. In common parlance, the notion of 'belonging to a set' predicates some property which is common to members of the set and members not in the set do not have the property. 'Belonging' to a set S can itself be considered to be the property which defines and is common to the members of S.
Further, the notion of an 'element' or 'point' P being a limit of a set S is that , no matter how small a ' neighborhood' of P we consider, there are infinitely many points of S in that neighbourhood. It is plausible to assume that if infinitely many elements close enough to P ( but possibly not all) have the property of 'belonging to S', P itself has the property of ' belonging to S'. In other words, we add as an additional Axiom, ( by restricting the full scope of the Axiom of Specification) that all subsets of a set are necessarily closed. My question is: Has any axiomatisation of set theory along these lines been attempted, and if so, any reference to it.