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Brouwer's fixed point theorem states that if a continuouos function $f$ maps a compact, convex set to itself, then $f$ has a fixed point in that set.

All these concepts are topological concepts, except convexity, which is a concept of affine spaces, since it relies on the "linear combination" of points.

So shouldn't we consider the theorem a theorem in $R^n$ euclidean analysis (or at least "affine analysis") rather than topology?

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    $\begingroup$ Convexity isn't needed per se. You need your space to be homeomorphic to a convex set (so homeomorphic to a disk is enough). Also, the proof often uses topological techniques. $\endgroup$ – Michael Burr May 9 '18 at 14:29
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As Lee's answer/the comment addresses, one generally proves Brouwer in a more general context using something called homology which lets one not speak of convexity at all. The theorem is sometimes phrased in terms of convexity to avoid having to define these broader conditions. But the theorem has nothing to do with convexity, rigidly construed.

Let $B$ be a compact, convex subset of $\mathbb{R}^n$, let $h: B \to \mathbb{R}^n$ be a homeomorphism onto its image, and let $f: h(B) \to h(B)$ continuous. Then $(h^{-1}\circ f \circ h): B \to B$ is continuous hence by the 'analytic' Brouwer fixed point theorem, there exists some $b^*$ such that $b^* = (h^{-1}\circ f \circ h)(b^*)$. But: $$ (h^{-1} \circ f)\big(h(b^*)\big) = (h^{-1})\big(h(b^*)\big) $$ hence $$ f(b^*) = b^* $$ by injectivity of $h$. Hence we can immediately drop 'convexity' for 'homeomorphic to a convex set,' and our condition already takes on a far more topological flavor.

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Topology is sometimes considered a branch of analysis. In fact, most concepts of topology are introduced first in an "advanced calculus" course, for instance concepts of distance, open balls, open sets, continuity of functions, and so on. Even Poincare's original terminology for topology, "analysis situs" sometimes translated as "analysis in the small", reflects these origins.

But topology diverges from analysis in several ways.

For example, analysis goes on to be concerned with concepts of infinitesimal change, reflected in the central notion of the derivative. And while derivatives are important in the branch of "differential topology" which is a kind of merging of analysis and topology, derivatives are not part of topology per se.

Another way topology diverges from analysis is in the way that its theorems are expressed. The modern topologist's statement of the Brouwer fixed point theorem is not expressed in terms of convexity, it is simply a theorem about a closed ball in $\mathbb{R}^n$:

  • If $B^n \subset \mathbb{R}^n$ is the closed unit ball in $\mathbb{R}^n$ and if $f : B^n \to B^n$ is continuous then there exists $p \in B^n$ such that $f(p)=p$.

In convex geometry one proves that every compact, convex subset of $\mathbb{R}^n$ is homeomorphic to the closed unit ball in $\mathbb{R}^k$ for some $k \le n$, and that's what you use to translate the convex geometer's version of the theorem into the modern topologist's version.

Perhaps the most characteristic way topology differs from analysis is in the tools that it employs. The modern proof of the Brouwer fixed point theorem uses singular homology theory, which is usually taught in a first year course of graduate topology. The 2-dimensional case of the theorem is sometimes taught first, because it can be done with a more elementary topological concept, namely the theory of the fundamental group, and can be done earlier in that graduate course.

If I had to summarize this more briefly: topology delves much more deeply into concepts related to continuity than analysis does, e.g. open sets and their properties; and topology employs certain tools of algebra much more commonly than analysis does, e.g. groups, modules, chain complexes. One other interesting feature of this is that since these topological tools were introduced, analysis has borrowed them and used them freely, for example concepts of cohomology theory.

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