Interlocking 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.^[1] To give the set S a topology means to say which subsets of S are "open", and to do so in a way that the following axioms are met:^[2]

1. The union of open sets is an open set.
2. The finite intersection of open sets is an open set.
3. S and the empty set ∅ are open sets.

Construction

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

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

The sets generated by X_n will be formed by all possible unions of finite intersections of the X_n.^[3]

References

1. Steen & Seebach (1978) pp.77 – 78
2. Steen & Seebach (1978) p.3
3. Steen & Seebach (1978) p.4

• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1978). Counterexamples in Topology (2nd ed.). Berlin, New York: Springer-Verlag. ISBN 3-540-90312-7. MR 507446. Zbl 0386.54001.