# Overlapping interval topology

In mathematics, the **overlapping interval topology** is a topology which is used to illustrate various topological principles.

## Definition

Given the closed interval of the real number line, the open sets of the topology are generated from the half-open intervals and with . The topology therefore consists of intervals of the form , , and with , together with itself and the empty set.

## Properties

Any two distinct points in
are topologically distinguishable under the overlapping interval topology as one can always find an open set containing one but not the other point. However, every non-empty open set contains the point 0 which can therefore not be separated from any other point in
, making
with the overlapping interval topology an example of a T_{0} space that is not a T_{1} space.

The overlapping interval topology is second countable, with a countable basis being given by the intervals
,
and
with
and *r* and *s* rational (and thus countable).

## See also

- Particular point topology, a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space

## References

- 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*(See example 53)*