Discrete space

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, 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, i.e., it defines all subsets as open sets. In particular, each singleton is an open set in the discrete topology.

Given a set X:

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

A metric space ${\displaystyle (E,d)}$ is said to be uniformly discrete if there exists a "packing radius" ${\displaystyle r>0}$ such that, for any ${\displaystyle x,y\in E}$, one has either ${\displaystyle x=y}$ or ${\displaystyle d(x,y)>r}$. 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.

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