Overlapping interval topology

In mathematics, the overlapping interval topology is a topology which is used to illustrate various topological principles.

Not to be confused with Interlocking interval topology.

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 ${\displaystyle a<0<b}$. The topology therefore consists of intervals of the form ${\displaystyle [-1,b)}$, ${\displaystyle (a,b)}$, and ${\displaystyle (a,1]}$ with ${\displaystyle a<0<b}$, 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 ${\displaystyle r<0<s}$ and r and s rational.

See also

• Particular point topology, a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space

References

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446 (See example 53)