Browder's Theorem: from One-Dimensional Parameter Space to General Parameter Space

2022/10/28 by Solan, Eilon, Solan, Omri Nisan
#54H25 #FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.2210.16369

Abstract

A parametric version of Brouwer's Fixed Point Theorem, which is proven using the fixed-point index, states that for every continuous mapping f : (X × Y) → Y, where X is nonempty, compact, and connected subset of a Hausdorff topological space and Y is a nonempty, convex, and compact subset of a locally-convex topological vector space, the set of fixed points of f, defined by C_f := \ (x,y) ∈ X × Y \colon f(x,y)=y\, has a connected component whose projection onto the first coordinate is X. In this note we provide an elementary proof for this result, using its reduction to the case X = [0,1].

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