The Sorgenfrey line is a nonstandard topology on the real line $\R$ . Its topology is defined by the following base of half open intervals$$ \mathcal{B} = \{ {[a,b)} \mid a,b\in\R, a<b \}.$$ Another name is lower limit topology, since a sequence $x_\alpha$ converges only if it converges in the standard topology and its limit is a limit from above (which, in this case, means that at most finitely many points of the sequence lie below the limit). For example, the sequence $(1/n)$ converges to $0$ , while $(-1/n)$ does not.

This topology is finer than the standard topology on $\R$ . The Sorgenfrey line is first countable and separable, but is not second countable. It is therefore not metrizable.

Bibliography

1

R. H. Sorgenfrey, On the topological product of paracompact spaces, Bulletin of the American Mathematical Society 53 (1947) 631-632. (This paper is available on-line from Project Euclid.)