# Which field of mathematics does Brouwer's Fixed Point Theorem belong to?

Forgive me for asking a seemingly stupid question: Is Brouwer's Fixed Point Theorem an Analysis theorem, or a Topology theorem?

It's talking about functions so I assume it's part of Analysis. But Wikipedia says it's a theorem in Topology.

A follow-up question is, if I want to study something like Brouwer's Theorem in future, could you suggest me a specific branch of mathematics? eg. a specific branch in Algebraic Topology or Functional Analysis, something like that. (I'm not sure if these fields are related to this theorem, I'm just trying to give examples)

p.s. I've just started my undergraduate year 2 and not even Point-Set Topology has been taught. So forgive me if it seems that I have a lack of mathematical knowledge.

• I would say it's quite clearly a topology theorem, but there are analytic proofs. I would say the question is not whether you want to study something like Brouwer's theorem, but rather what proof of it you find interesting, and then study something lile that. Commented Oct 9, 2022 at 10:30
• It depends how you look at it, if you only consider the function on a disk, and you state it and prove it by analytic means, that would make it a theorem in Analysis. However, it can be viewed and generalised in topological terms only. In particular, the theorem depends on the topological shape of the domain of the function, it is true for disks, but false for rings/tori. I guess the problem is that your question is too vague - what makes a theorem strictly analytic or topological? These areas of mathematics have many intersections, but if I had to choose between the two I'd say topological. Commented Oct 9, 2022 at 10:32
• Continuity and compactness are core topics in topology. I think it firmly belongs in that camp. Commented Oct 9, 2022 at 10:56
• When I read the title, I imagined the dispute was between "point-set topology" and "algebraic topology". Commented Oct 9, 2022 at 11:13
• You need some topology to understand the underpinnings of analysis, and you need some analysis at least to motivate some parts of topology. So take both. Take lots of math and see what other results and proofs appeal to you: it's too early for you to specialize. Commented Oct 9, 2022 at 16:22