discrete space

The discrete topology on a set X is the topologyMathworldPlanetmathPlanetmath given by the power setMathworldPlanetmath of X. That is, every subset of X is open in the discrete topology. A space equipped with the discrete topology is called a discrete space.

The discrete topology is the http://planetmath.org/node/3290finest topology one can give to a set. Any set with the discrete topology is metrizable by defining d(x,y)=1 for any x,yX with xy, and d(x,x)=0 for any xX.

The following conditions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    X is a discrete space.

  2. 2.

    Every singleton in X is an open set.

  3. 3.

    Every subset of X containing x is a neighborhoodMathworldPlanetmathPlanetmath of x.

Note that any bijection between discrete spaces is trivially a homeomorphismPlanetmathPlanetmath.

Discrete Subspaces

If Y is a subset of X, and the subspace topology on Y is discrete, then Y is called a discrete subspace or discrete subset of X.

Suppose X is a topological space and Y is a subset of X. Then Y is a discrete subspace if and only if, for any yY, there is an open SX such that



  1. 1.

    , as a metric space with the standard distance metric d(m,n)=|m-n|, has the discrete topology.

  2. 2.

    , as a subspaceMathworldPlanetmath of or with the usual topology, is discrete. But , as a subspace of or with the trivial topology, is not discrete.

  3. 3.

    , as a subspace of with the usual topology, is not discrete: any open set containing q contains the intersectionMathworldPlanetmath U=B(q,ϵ) of an open ball around q with the rationals. By the Archimedean property, there’s a rational numberPlanetmathPlanetmath between q and q+ϵ in U. So U can’t contain just q: singletons can’t be open.

  4. 4.

    The set of unit fractions F={1/nn}, as a subspace of with the usual topology, is discrete. But F{0} is not, since any open set containing 0 contains some unit fraction.

  5. 5.

    The productPlanetmathPlanetmathPlanetmath of two discrete spaces is discrete under the product topology. The product of an infiniteMathworldPlanetmath number of discrete spaces is discrete under the box topology, but if an infinite number of the spaces have more than one elementMathworldMathworld, it is not discrete under the product topology.

Title discrete space
Canonical name DiscreteSpace
Date of creation 2013-03-22 12:29:56
Last modified on 2013-03-22 12:29:56
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 17
Author mathcam (2727)
Entry type Definition
Classification msc 54-00
Synonym discrete topological space
Related topic Discrete2
Defines discrete subspace
Defines discrete topology
Defines discrete space
Defines discrete subset