Interlocking interval topology
# 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:

* The union of open sets is an open set.
* The finite intersection of open sets is an open set.
* 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

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

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

