Sorgenfrey line

The Sorgenfrey line is a nonstandard topology on the real line $ℝ$.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 isa limit from above (which, in this case, means that at most finitely manypoints 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.