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] 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.
Not to be confused with Overlapping interval topology.

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.[3]

gollark: Current physical evidence is overwhelmingly in favour of it being globey. That doesn't mean that we have *proven* it must be a globe.
gollark: ... no, it's shown that *in our physical models*, this is the case, and I think in some cases they just start from that as an assumption.
gollark: It *cannot be proven* that this holds in all situations ever, because this is a statement about reality and not our models.
gollark: As far as anyone knows, yes.
gollark: Current physical theories say it can't. They seem to be right about this so far, but the models *do not create reality*, it goes the other way round.

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