I could write a short essay in response...
TLDR is if put sets first, since that stuff is foundational to the rest. Topology has a formal math definition which may surprise you, not exactly what you'd expect, just a few rules about sets inside a larger space, and continuous functions types of things.
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i think that topology leads to sets, by the categorisation, but sets don't exist until you have a variety of elements with uniqueness first. and geometry goes second because you use geometry to build the phylogeny of sets. so maybe topology -> sets -> geometry
like, you can't have geometry without polyhedra, or vectors, but you can have topology without sets. a circle is not a set, it is an element of a set. same can be said of a plane or a bounded surface. they form into a set of sets, but they precede the existence of sets.
so i still say that topology is the root. topology delineates the form from the mold, the day from the night.