# 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 ${\displaystyle [-1,1]}$  of the real number line, the open sets of the topology are generated from the half-open intervals ${\displaystyle [-1,b)}$  and ${\displaystyle (a,1]}$  with ${\displaystyle a<0 . The topology therefore consists of intervals of the form ${\displaystyle [-1,b)}$ , ${\displaystyle (a,b)}$ , and ${\displaystyle (a,1]}$  with ${\displaystyle a<0 , together with ${\displaystyle [-1,1]}$  itself and the empty set.

## Properties

Any two distinct points in ${\displaystyle [-1,1]}$  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 ${\displaystyle [-1,1]}$ , making ${\displaystyle [-1,1]}$  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 ${\displaystyle [-1,s)}$ , ${\displaystyle (r,s)}$  and ${\displaystyle (r,1]}$  with ${\displaystyle r<0  and r and s rational (and thus countable).