The Hausdorff topology as a moduli space Article

Gillam, WD, Karan, A. (2017). The Hausdorff topology as a moduli space . TOPOLOGY AND ITS APPLICATIONS, 232 102-111. 10.1016/j.topol.2017.10.003

cited authors

  • Gillam, WD; Karan, A



  • 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.

publication date

  • December 1, 2017

published in

Digital Object Identifier (DOI)

start page

  • 102

end page

  • 111


  • 232