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 $[-1,1]$ of the real number line, the open sets of the topology are given by $[-1,b)$ for $b>0$ and $(a,1]$ for $a<0$. Note that sets of the form $(a,b)$, with $b>0$ and $a<0$, are also open.

Properties

The overlapping interval topology has various properties, including:

• It is an example of T0 space that is not a T1 space, since the point 0 is not closed.

• It 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

• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology, (1978) Dover Publications, ISBN 0-486-68735-X. (See example 53)