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]

The union of open sets is an open set.
The finite intersection of open sets is an open set.
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}:=\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 intersections of finite unions of the X_n.^[3]

References

^ Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, pp. 77–78, ISBN 048668735X
^ Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, p. 3, ISBN 048668735X
^ Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, p. 4, ISBN 048668735X

[1] Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, pp. 77–78, ISBN 048668735X

[2] Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, p. 3, ISBN 048668735X

[3] Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, p. 4, ISBN 048668735X

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