# Sorgenfrey line

The Sorgenfrey line is a nonstandard topology on the real line $\mathbb{R}$. Its topology is defined by the following base of half open intervals

 $\mathcal{B}=\{{[a,b)}\mid a,b\in\mathbb{R},a

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 $\mathbb{R}$. The Sorgenfrey line is first countable and separable, but is not second countable. It is therefore not metrizable.

## References

Title Sorgenfrey line SorgenfreyLine 2013-03-22 13:03:45 2013-03-22 13:03:45 yark (2760) yark (2760) 9 yark (2760) Example msc 55-00 msc 54-00 msc 22-00 Sorgenfrey topology lower limit topology