Discrete Space - Definitions

Definitions

Given a set X:

• the discrete topology on X is defined by letting every subset of X be open (and hence also closed), and X is a discrete topological space if it is equipped with its discrete topology;
• the discrete uniformity on X is defined by letting every superset of the diagonal {(x,x) : x is in X} in X × X be an entourage, and X is a discrete uniform space if it is equipped with its discrete uniformity.
• the discrete metric on X is defined by

for any . In this case is called a discrete metric space or a space of isolated points.

• a set S is discrete in a metric space, for, if for every, there exists some (depending on ) such that for all ; such a set consists of isolated points. A set S is uniformly discrete in the metric space, for, if there exists ε > 0 such that for any two distinct, > ε.

A metric space is said to be uniformly discrete if there exists such that, for any, one has either or . 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.