View source for Rational sequence topology

{{short description|Mathematical theory related to general topology}}

In [[mathematics]], more specifically [[general topology]], the '''rational sequence topology''' is an example of a [[Topological space#topology|topology]] given to the set '''R''' of [[real number]]s.

== Construction ==

For each [[irrational number]] ''x'' take a [[sequence]] of [[rational number]]s {''x<sub>k</sub>''} with the property that {''x<sub>k</sub>''} [[Limit of a sequence|converge]]s to ''x'' with respect to the [[Euclidean topology]].

The '''rational sequence topology'''<ref>{{Citation|first=L. A.|last=Steen|first2=J. A.|last2=Seebach|title=[[Counterexamples in Topology]]|publisher=Dover|year=1995|page=87|ISBN=0-486-68735-X}}</ref> is specified by letting each rational number [[singleton (mathematics)|singleton]] to be open, and using as a [[neighborhood base]] for each irrational number ''x'', the sets <math> U_n(x) = \{ x_k : k \ge n \} \cup \{x\}.</math>

==References==

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[[Category:Topological spaces]]