In this way, topological spaces made it possible to explore novel parts of the mathematical landscape. For instance, they provided a new, distance-free way of understanding concepts like continuity and connectedness, which is a profoundly powerful and counterintuitive capability to have. This has allowed mathematicians to generalize those ideas to a much broader range of settings — and to prove important statements in a wide variety of fields. For example, results like the fundamental theorem of algebra, whose proof tripped up even mathematical giants like Carl Friedrich Gauss, end up with very simple proofs once topological arguments come into play.
https://www.quantamagazine.org/two-researchers-are-rebuilding-mathematics-from-the-ground-up-20260520/
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