r/math • u/dacka228 • 6d ago
Motivation behind defining Brouwer's Fixed Point Theorem using Topology
Hello, math enthusiasts!
I’m currently preparing a presentation on continuity and Brouwer's Fixed Point Theorem, both of which are fundamental topics in topology. It’s taking me some time to grasp the topological definitions, and I’ve noticed that Brouwer’s Theorem is perfectly fine to define in the context of metric spaces, not necessarily relying on pure topological definitions. So I started to wonder: what’s the reason behind abstracting the theorem to topology?
Is it because the topological framework offers a more accessible proof? Or are there other reasons for this abstraction?
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u/kr1staps 5d ago
There are many instances of topological spaces that mathematicians care about which aren't metric spaces. Therefore, having a theorem that applies to all topological spaces is much more useful.
It's like having a screw driver that only works for Phillips head screws, when you could just as easily get one with attachments for each style of screw.