Discrete space

In topology, a discrete space is a topological space with the discrete topology, in which every subset is open.

Properties

• A discrete space is metrizable, with the topology induced by the discrete metric.
• A discrete space is a uniform space with the discrete uniformity.
• A discrete space is compact if and only if it is finite.
• A discrete space is connected if and only if it has at most one point.
• Every map from a discrete space to a topological space is continuous.

References

• Lynn Arthur Steen; J. Arthur Seebach jr (1978). Counterexamples in Topology. Berlin, New York: Springer-Verlag, 41-42. ISBN 0-387-90312-7.