2012/09/12 by Michael Shulman, Shulman, Michael
Computer Science · Mathematics · #Advanced Topology and Set Theory #Category Theory (math.CT) #Computability, Logic, AI Algorithms #FOS: Mathematics #General Topology (math.GN) #Homotopy and Cohomology in Algebraic Topology #math.CT #math.GN
paper · pdf · doi:10.48550/arxiv.1209.2735
31 pages; plenty of exercises
arxiv created 2012/09/12 · openalex publication_date 2012/09/12 · arxiv updated 2012/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In these expository notes, intended for students without background in point-set topology, we develop the basic theory of the Stone-Cech compactification without reference to open sets, closed sets, filters, or nets. In particular, this means we cannot use any of the usual definitions of topological space. This may seem like proposing to run a marathon while hopping on one foot, but it is easier than it may appear, and not devoid of interest. We use gauge spaces (uniform spaces presented by a family of pseudometrics); we define compactness as total boundedness plus completeness; and we define completeness using a variation on Lawvere's categorical characterization of completeness for metric spaces.