Zero-dimensional space

Summary

In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space.[1][2] A graphical illustration of a nildimensional space is a point.[3]

DefinitionEdit

Specifically:

  • A topological space is zero-dimensional with respect to the Lebesgue covering dimension if every open cover of the space has a refinement which is a cover by disjoint open sets.
  • A topological space is zero-dimensional with respect to the finite-to-finite covering dimension if every finite open cover of the space has a refinement that is a finite open cover such that any point in the space is contained in exactly one open set of this refinement.
  • A topological space is zero-dimensional with respect to the small inductive dimension if it has a base consisting of clopen sets.

The three notions above agree for separable, metrisable spaces.[citation needed][clarification needed]

Properties of spaces with small inductive dimension zeroEdit

HypersphereEdit

The zero-dimensional hypersphere is a pair of points. The zero-dimensional ball is a point.

NotesEdit

  • Arhangel'skii, Alexander; Tkachenko, Mikhail (2008). Topological Groups and Related Structures. Atlantis Studies in Mathematics. Vol. 1. Atlantis Press. ISBN 978-90-78677-06-2.
  • Engelking, Ryszard (1977). General Topology. PWN, Warsaw.
  • Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.

ReferencesEdit

  1. ^ "zero dimensional". planetmath.org. Retrieved 2015-06-06.
  2. ^ Hazewinkel, Michiel (1989). Encyclopaedia of Mathematics, Volume 3. Kluwer Academic Publishers. p. 190. ISBN 9789400959941.
  3. ^ Wolcott, Luke; McTernan, Elizabeth (2012). "Imagining Negative-Dimensional Space" (PDF). In Bosch, Robert; McKenna, Douglas; Sarhangi, Reza (eds.). Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture. Phoenix, Arizona, USA: Tessellations Publishing. pp. 637–642. ISBN 978-1-938664-00-7. ISSN 1099-6702. Retrieved 10 July 2015.