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

Properties

Any two distinct points in [math]\displaystyle{ [-1,1] }[/math] 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 [math]\displaystyle{ [-1,1] }[/math], making [math]\displaystyle{ [-1,1] }[/math] 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 [math]\displaystyle{ [-1,s) }[/math], [math]\displaystyle{ (r,s) }[/math] and [math]\displaystyle{ (r,1] }[/math] with [math]\displaystyle{ r \lt 0 \lt s }[/math] and r and s rational.

See also

• List of topologies
• 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  (See example 53)