2016/01/11 by W. D. Gillam, Gillam, W. D., A. Karan +1
Mathematics · #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AG
paper · pdf · doi:10.48550/arxiv.1601.02425
arxiv created 2016/01/11 · openalex publication_date 2016/01/11 · arxiv updated 2016/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 1914, F. Hausdorff defined a metric on the set of closed subsets of a metric space X. This metric induces a topology on the set H of compact subsets of X, called the Hausdorff topology. We show that the topological space H represents the functor on the category of sequential topological spaces taking T to the set of closed subspaces Z of T × X for which the projection π1 : Z → T is open and proper. In particular, the Hausdorff topology on H depends on the metric space X only through the underlying topological space of X. The Hausdorff space H provides an analog of the Hilbert scheme in topology. As an example application, we explore a certain quotient construction, called the Hausdorff quotient, which is the analog of the Hilbert quotient in algebraic geometry.