# 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.[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

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

## References

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