# 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 $[-1,b)$  and $(a,1]$  with $a<0 . The topology therefore consists of intervals of the form $[-1,b)$ , $(a,b)$ , and $(a,1]$  with $a<0 , 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  and r and s rational (and thus countable).