Overlapping interval topology

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

Given the closed interval

of the real number line, the open sets of the topology are generated from the half-open intervals

The topology therefore consists of intervals of the form

itself and the empty set.

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 T0 space that is not a T1 space.

The overlapping interval topology is second countable, with a countable basis being given by the intervals