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Andrés E. Caicedo
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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?