Given the historical development of modern mathematics, everything is ultimately encoded as a set (possibly with some additional structure, also encoded as set(s) ). For example, a topological space is an ordered pair $(X, \tau)$ where $X$ is the underlying point-set and $\tau$ is a set of subsets on $X$ called it's topology, obeying some axioms. But what if I don't care about sets in particular, especially just the sets of ZFC, what if I want to study mathematical structures with notions of nearness, connectedness, continuity, etc. and do not want to be constrained to an underlying set theoretic encoding? Does category theory offer me a way to study the "essence" of a topological space that will hold no matter what encoding (i.e. foundational theory) I use?



Topological space (or math structure more generally) without encoding as set
There are several ways to understand this question, leading to different answers.


1. If the question is about studying *topological spaces*, in the sense of points equipped with open sets, independent of a particular axiom system for set theory, then the commentators are right: ordinary point-set topology is (like nearly all of mathematics) already independent of any particular axiom system for set theory.
2. If the question is about studying things that behave like topological spaces but don't have a "set of points", then the other commentators are right that you should look into locales and topoi.
3. If the question is about "notions of nearness, connectedness, continuity" in a more general setting that are *not* necessarily defined in terms of a notion of "open set", then you may be interested in categorical topology (e.g. [topological categories](https://ncatlab.org/nlab/show/topological+concrete+category)) or [axiomatic cohesion](https://ncatlab.org/nlab/show/cohesive+topos).


