2019/08/27 by Jerzy Dydak, Dydak, Jerzy
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #General Topology (math.GN) #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Metric Geometry (math.MG) #Primary 54F45 #Secondary 55M10
paper · pdf · doi:10.48550/arxiv.1908.09986
openalex publication_date 2019/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present an idea of unifying small scale (topology, proximity spaces, uniform spaces) and large scale (coarse spaces, large scale spaces). It relies on an analog of multilinear forms from Linear Algebra. Each form has a large scale compactification and those include all well-known compactifications: Higson corona, Gromov boundary of hyperbolic spaces, the visual boundary of CAT(0)-spaces, \v Cech-Stone compactification, Samuel-Smirnov compactification, and Freudenthal compactification. As an application we get simple proofs of results generalizing well-known theorems from coarse topology. A new result (at least to the author) is the following (see \refHomeoOfHigsonImpliesLSEquivalence): A coarse bornologous function f:X→ Y of metrizable large scale spaces is a large scale equivalence if and only if it induces a homeomorphism of Higson coronas. This paper is an extension of \citeJD2 and, at the same time, it overrides \citeJD2.