2010/10/12 by Robert Cauty, Cauty, Robert
Mathematics · #54C55 #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #General Topology (math.GN) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1010.2401
openalex publication_date 2010/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A topological space is locally equiconnected if there exists a neighborhood U of the diagonal in X× X and a continuous map λ:U×[0,1]→ X such that λ(x,y,0)=x, λ(x,y,1)=y et λ(x,x,t)=x for (x,y)∈ U and (x,t)∈ X×[0,1]. This class contains all ANRs, all locally contractible topological groups and the open subsets of convex subsets of linear topological spaces. In a series of papers, we extended the fixed point theory of compact continuous maps, which was well developped for ANRs, to all separeted locally equiconnected spaces. This generalization includes a proof of Schauder's conjecture for compact maps of convex sets. This paper is a survey of that work. The generalization has two steps: the metrizable case, and the passage from the metrizable case to the general case. The metrizable case is, by far, the most difficult. To treat this case, we introduced in [4] the notion of algebraic ANR. Since the proof that metrizable locally equiconnected spaces are algebraic ANRs is rather difficult, we give here a detaled sketch of it in the case of a compact convex subset of a metrizable t.v.s.. The passage from the metrizable case to the general case uses a free functor and representations of compact spaces as inverse limits of some special inverse systems of metrizable compacta.