Overlapping interval topology

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


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.


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 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


  • 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)
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