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.[1]

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 := ( 0 , 1 n ) ∪ ( n , n + 1 ) = { x ∈ R + : 0 < x < 1 n    or    n < x < n + 1 } . {\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 Xn will be formed by all possible unions of finite intersections of the Xn.[2]

See also

  • List of topologies

References

  1. ^ Steen & Seebach (1978) pp.77 – 78
  2. ^ Steen & Seebach (1978) p.4
  • Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1978). Counterexamples in Topology (2nd ed.). Berlin, New York: Springer-Verlag. ISBN 3-540-90312-7. MR 0507446. Zbl 0386.54001.