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, MR 507446 (See example 53)