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]

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.^[2]

See also

• List of topologies