A discrete space is simply a topological space equipped with the discrete topology. A discrete space is always a metric space, namely the metric space with the same underlying set and with the discrete metric.
Discrete Space
See also
Discrete Topology, Metric Space, Topological SpaceThis entry contributed by Rasmus Hedegaard
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Hedegaard, Rasmus. "Discrete Space." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/DiscreteSpace.html