Overlapping interval topology

Given the closed interval [-1,1] of the real number line, the open sets of the topology are generated from the half-open intervals (a,1] with a and [-1,b) with b > 0. The topology therefore consists of intervals of the form [-1,b), (a,b), and (a,1] with a , together with [-1,1] itself and the empty set. ==Properties==

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 and r and s rational. == See also ==