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DEFINITIONS

Given a set X:
  • the discrete topology on X is defined by letting every Subset of X be Open , and X is a '''discrete topological space''' if it is equipped with its discrete topology;

  • the discrete , and X is a discrete uniform space''' if it is equipped with its discrete uniformity.

  • the discrete Metric on X is defined by letting the distance between any Distinct points x and y be , and X is a '''discrete metric space''' if it is equipped with its discrete metric.


A metric space ( E , d ) is said to be ''uniformly discrete'' if there exists r>0 such that, for any x,y \in E, one has either x=y or d(x,y)>r. The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set {1, 1/2, 1/4, 1/8, ...} of real numbers.


PROPERTIES

The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology.
Thus, the different notions of discrete space are compatible with one another.