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I'm confused as to how to develop intuition for the "classical" results of topology. By these, I mean results like Brouwer's fixed point theorem, the hairy-ball theorem, Borsuk-Ulam theorem, fundamental theorem of algebra, etc., things that can be proven with knowledge of the fundamental group of the circle.

I'm currently reading Chapter 9 of Munkres' Topology, and I can follow the arguments presented in the text. However, I don't feel as though I'm understanding them intuitively. Is there a reference that could act as a supplement to this chapter that treats these "classic" theorems in more depth with emphasis on intuitive understanding? How did you guys develop this intuition and begin "thinking like a topologist"?

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  • $\begingroup$ It's worth seeking out the Sperner's Lemma proof of Brouwer's Fixed Point theorem. It is constructive. The fundamental theorem of algebra also has a constructive proof, I believe. But I've never seen a truly "intuitive" proof of these theorems, or an intuition for why they are true. $\endgroup$ – Thomas Andrews Jul 11 '17 at 18:15
  • $\begingroup$ Well recently, I watched one of 3Blue1Brown's topology videos where he explained the Borsuk-Ulam theorem rather intuitively. Watching that made me realize I don't really understand any of these theorems at all! So I was hoping that there would be similarly intuitive explanations of the other "classic" results. $\endgroup$ – pianyon Jul 11 '17 at 18:18
  • $\begingroup$ Remarkably, the fundamental group might be the "intuitive" proof of these. When I think of FTA, I think of the homotopy proof for my intuition. There is a lot of complicated technical homotopy stuff used to prove the FTA with homotopy, but the idea that a polynomial sends one loop around zero to a contractable loop in $\mathbb C\setminus\{0\}$ and another loop around zero does not is fairly intuitive. $\endgroup$ – Thomas Andrews Jul 11 '17 at 18:20
  • $\begingroup$ For the fundamental theorem of algebra, I would recommend Abel's Theorem in Problems and Solutions. As for the intuition of fixed-point theorems in general, would it help to ask: Which transformations certainly fail to have a fixed point, and why? An example of transformations that are kind of "opposite" to having a fixed point is ergodic transformations (also look at mixing transformations). $\endgroup$ – avs Jul 11 '17 at 18:22
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    $\begingroup$ I felt I didn't understand topology until I learned differential topology. I feel that a topological space is not something to get inuition about. The objects are to general and one would get the feeling that topology is the study of things like the separation axioms ( find a space that is T4 but not Tpi). Have a look at guillemin and pollack or milnors topology from the differential viewpoint. $\endgroup$ – Thomas Rot Jul 11 '17 at 18:27

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