Overlapping interval topology
Given the closed interval of the real number line, the open sets of the topology are generated from the half-open intervals and with . The topology therefore consists of intervals of the form , , and with , together with 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 , and with and r and s rational (and thus countable).