Interlocking interval topology

Not to be confused with Overlapping interval topology.

In mathematics, and especially general topology, the interlocking interval topology is an example of a topology on the set S := R⁺ \ Z⁺, i.e. the set of all positive real numbers that are not positive whole numbers.

Construction

The open sets in this topology are taken to be the whole set S, the empty set ∅, and the sets generated by

$X_{n}:=\left(0,{\frac {1}{n}}\right)\cup (n,n+1)=\left\{x\in {\mathbf {R} }^{+}:0<x<{\frac {1}{n}}\ {\text{ or }}\ n<x<n+1\right\}.$

The sets generated by X_n will be formed by all possible unions of finite intersections of the X_n.

See also

• List of topologies

References

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1978). Counterexamples in Topology (2nd ed.). Berlin, New York: Springer-Verlag. ISBN 3-540-90312-7. MR 0507446. Zbl 0386.54001.