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 formula_1 of the real number line, the open sets of the topology are generated from the half-open intervals formula_2 and formula_3 with formula_4. The topology therefore consists of intervals of the form formula_2, formula_6, and formula_3 with formula_4, together with formula_1 itself and the empty set.
Properties.
Any two distinct points in formula_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 formula_1, making formula_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 formula_13, formula_14 and formula_15 with formula_16 and "r" and "s" rational.