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. 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:

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

[math]\displaystyle{ X_n := \left(0,\frac{1}{n}\right) \cup (n,n+1) = \left\{ x \in {\mathbf R}^+ : 0 \lt x \lt \frac{1}{n} \ \text{ or } \ n \lt x \lt n+1 \right\}. }[/math]

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.