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

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.