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 $$[-1,1]$$ of the real number line, the open sets of the topology are generated from the half-open intervals $$(a,1]$$ with $$a < 0$$ and $$[-1,b)$$ with $$b > 0$$. 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.

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 $$r < 0 < s $$ and r and s rational.