Interlocking interval topology

In mathematics, and especially general topology, the interlocking interval topology is an example of a topology on the set , 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:
Construction.
The open sets in this topology are taken to be the whole set "S", the empty set ∅, and the sets generated by
The sets generated by "Xn" will be formed by all possible unions of finite intersections of the "Xn".