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Concept# Discrete space

Summary

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are isolated from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology.
Definitions
Given a set X:
A metric space (E,d) is said to be uniformly discrete if there exists a 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 \left{2^{-n} : n \in \N_0\right}.
Properties
The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology o

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