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.
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Ah, I see how you're viewing things somewhat. Maybe based on intuitive notions of topology that's true, and historically that may be how things evolved. Set theory is now considered the foundational math, though it wasn't formalized until "modern" times when troubles began to arise in foundations of the existing math (as they sorta pushed its limits I guess).
Now the view is you start with set theory, build out some concepts and definitions, and other stuff is more special areas of interest within that framework.
Look up how topology is defined axiomatically and you'll likely be surprised at how different it is from what you would expect. Just a collection of sets and a couple rules about intersections and unions, that's it. Similarly with algebra. It's all sort of a modern movement from late 1800s onward toward more formalization and abstraction. Type of stuff I'm reading about now.
