# Discrete space

From Citizendium

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.