Discrete space

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In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.

Contents

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 ρ on X is defined by

for any x,y \in X. In this case (X,ρ) is called a discrete metric space or a space of isolated points.

  • a set S is discrete in a metric space (X,ρ), for S \subseteq X, if for any two distinct x, y \in S, ρ(x,y) > 0; such a set consists of isolated points. A set S is uniformly discrete in the metric space (X,ρ), for S \subseteq X, if there exists ε > 0 such that for any two distinct x, y \in S, ρ(x,y) > ε.

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. On the other hand, the underlying topology of a non-discrete uniform or metric space can be discrete; an example is the metric space X := {1/n : n = 1,2,3,...} (with metric inherited from the real line and given by d(x,y) = |x − y|). Obviously, this is not the discrete metric; also, this space is not complete and hence not discrete as a uniform space. Nevertheless, it is discrete as a topological space. We say that X is topologically discrete but not uniformly discrete or metrically discrete.

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