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 with formula_3 and formula_4 with formula_5. The topology therefore consists of intervals of the form formula_4, formula_7, and formula_2 with formula_9, 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 T space that is not a T space. The overlapping interval topology is second countable, with a countable basis being given by the intervals formula_14, formula_15 and formula_16 with formula_17 and "r" and "s" rational.