Overlapping interval topology

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

DefinitionEdit

Given the closed interval   of the real number line, the open sets of the topology are generated from the half-open intervals   and   with  . The topology therefore consists of intervals of the form  ,  , and   with  , together with   itself and the empty set.

PropertiesEdit

Any two distinct points in   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  , making   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  ,   and   with   and r and s rational (and thus countable).

See alsoEdit

  • 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

ReferencesEdit

  • 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)