# Discrete topology

From Encyclopedia of Mathematics

*on a set $X$*

The topology in which every set is open (and therefore every set is closed). The discrete topology is the largest element in the lattice of all topologies on the given set. If $X$ is a space with the discrete topology, then every map from $X$ to any other topological space is continuous.

#### References

- J.L. Kelley, "General topology", Graduate Texts in Mathematics
**27**Springer (1975) ISBN 0-387-90125-6 MR0370454 Zbl 0306.54002

**How to Cite This Entry:**

Discrete topology.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Discrete_topology&oldid=37244

This article was adapted from an original article by A.A. Mal'tsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article