In mathematics, the **partition topology** is a topology that can be induced on any set *X* by partitioning *X* into disjoint subsets *P*; these subsets form the basis for the topology. There are two important examples which have their own names:

- The
**odd–even topology**is the topology where and - The
**deleted integer topology**is defined by letting and .

The trivial partitions yield the discrete topology (each point of *X* is a set in *P*) or indiscrete topology ().

Any set *X* with a partition topology generated by a partition *P* can be viewed as a pseudometric space with a pseudometric given by:

This is not a metric unless *P* yields the discrete topology.

The partition topology provides an important example of the independence of various separation axioms. Unless *P* is trivial, at least one set in *P* contains more than one point, and the elements of this set are topologically indistinguishable: the topology does not separate points. Hence *X* is not a Kolmogorov space, nor a T_{1} space, a Hausdorff space or an Urysohn space. In a partition topology the complement of every open set is also open, and therefore a set is open if and only if it is closed. Therefore, *X* is regular, completely regular, normal and completely normal. *X/P* is the discrete topology.

## References

This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. (April 2020) (Learn how and when to remove this template message) |

- Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978],
*Counterexamples in Topology*(Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446 CS1 maint: discouraged parameter (link)