Rational sequence topology

In mathematics, more specifically general topology, the rational sequence topology is an example of a topology given to the set R of real numbers.

Construction

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For each irrational number x take a sequence of rational numbers {xk} with the property that {xk} converges to x with respect to the Euclidean topology.

The rational sequence topology[1] is specified by letting each rational number singleton to be open, and using as a neighborhood base for each irrational number x, the sets  

References

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  1. ^ Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, p. 87, ISBN 0-486-68735-X