In mathematics, the overlapping interval topology is a topology which is used to illustrate various topological principles.
Given the closed interval [ − 1 , 1 ] of the real number line, the open sets of the topology are generated from the half-open intervals [ − 1 , b ) and ( a , 1 ] with a < 0 < b . The topology therefore consists of intervals of the form [ − 1 , b ) , ( a , b ) , and ( a , 1 ] with a < 0 < b , together with [ − 1 , 1 ] itself and the empty set.
Any two distinct points in [ − 1 , 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 [ − 1 , 1 ] , making [ − 1 , 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 [ − 1 , s ) , ( r , s ) and ( r , 1 ] with r < 0 < s and r and s rational (and thus countable).
