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  of the real number line, the open sets of the topology are generated from the half-open intervals  with  and   with . The topology therefore consists of intervals of the form , , and  with , together with  itself and the empty set. Properties
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. See also 

 List of topologies
 Particular point topology, a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space

References
  (See example 53)  

Topological spaces