SorgenfreyLine 2002-09-21 21:39:53 2007-08-04 15:23:07 Example topological space lower limit topology \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \def\sse{\subseteq} \def\bigtimes{\mathop{\mbox{\Huge $\times$}}} \def\impl{\Rightarrow} \def\R{\mathbb{R}} The \emph{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 \emph{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. \begin{thebibliography}{9} \bibitem{sorgenfrey} R.~H.~Sorgenfrey, {\it On the topological product of paracompact spaces}, Bulletin of the American Mathematical Society 53 (1947) 631--632. (This paper is \PMlinkexternal{available on-line}{http://projecteuclid.org/euclid.bams/1183510809} from Project Euclid.) \end{thebibliography}