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Interlocking interval topology

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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:

  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.

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 } .

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

References

Interlocking interval topology Wikipedia
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