zero dimensional

Definition 1.

[1, 2] Suppose X is a topological spaceMathworldPlanetmath. If X has a basis consising of clopen sets, then X is said to be .

Examples of zero-dimensional spaces are: the set of rational numbersPlanetmathPlanetmathPlanetmath (with subspace topology induced from the usual metric topology on , the set of real numbers), the Cantor space, as well as the Sorgenfrey line.

The concepts of zero-dimentionality and total disconnectedness are closely related. Indeed, every zero-dimentional T1 space ( is totally disconnected. Furthermore, if a topological space is locally compact and HausdorffPlanetmathPlanetmath, then the notions of zero-dimentionality and total disconnectedness are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.


  • 1 L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., 1970.
  • 2 S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.
Title zero dimensional
Canonical name ZeroDimensional
Date of creation 2013-03-22 14:41:05
Last modified on 2013-03-22 14:41:05
Owner matte (1858)
Last modified by matte (1858)
Numerical id 9
Author matte (1858)
Entry type Definition
Classification msc 54-00
Synonym zero-dimensional
Related topic SeparationAxioms