Topological space (or math structure more generally) without encoding as set 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?
 A: *

*Some categories are not concrete, i.e. one cannot think to objects as sets and to morphisms as function. It's the case of homotopy category of topological space.

*Sober spaces are equivalent to localic toposes with enough points but honestly, I am not sure this is so far from a set theoretic approach.
A: There are several ways to understand this question, leading to different answers.


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*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.

*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.

*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) or axiomatic cohesion.
