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

Properties

Any two distinct points in [ − 1 , 1 ] {\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 [ − 1 , 1 ] {\displaystyle [-1,1]} , making [ − 1 , 1 ] {\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 [ − 1 , s ) {\displaystyle [-1,s)} , ( r , s ) {\displaystyle (r,s)} and ( r , 1 ] {\displaystyle (r,1]} with r < 0 < s {\displaystyle r<0<s} 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, MR 0507446 (See example 53)