Rational sequence topology

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

To give R a topology means to say which subsets of R are "open", and to do so in a way that the following axioms are met:^[1]

1. The union of open sets is an open set.
2. The finite intersection of open sets is an open set.
3. R and the empty set ∅ are open sets.

Construction

Let x be an irrational number (cf. rational number). Take a sequence of rational numbers {x_k} with the property that {x_k} converges, with respect to the Euclidean topology, towards x as k tends towards infinity. Informally, this means that each of the numbers in the sequence get closer and closer to x as we progress further and further along the sequence.

The rational sequence topology is given by defining both the whole set R and the empty set ∅ to be open, defining each rational number singleton to be open, and using as a basis for the irrational number x, the sets ^[2]

$U_n(x) := \{ x_k : n \le k < \infty \} \cup \{x\} .$

References

1. Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, p. 3, ISBN 0-486-68735-X
2. Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, p. 87, ISBN 0-486-68735-X