Overlapping interval topology

Not to be confused with Interlocking interval topology.

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

Definition

Given the closed interval [ − 1,1] of the real number line, the open sets of the topology are generated from the half-open intervals [ − 1,b) and (a,1] with a < 0 < b. The topology therefore consists of intervals of the form [ − 1,b), (a,b), and (a,1] with a < 0 < b, together with [ − 1,1] itself and the empty set.

Properties

Any two distinct points in [ − 1,1] 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 [ − 1,1], making [ − 1,1] with the overlapping interval topology an example of a T0 space that is not a T1 space.

The overlapping interval topology is second countable, with a countable basis being given by the intervals [ − 1,s), (r,s) and (r,1] with r < 0 < s and r and s rational (and thus countable).

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, MR507446  (See example 53)