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

References

1 R. H. Sorgenfrey, On the topological product of paracompact spaces, Bulletin of the American Mathematical Society 53 (1947) 631–632. (This paper is http://projecteuclid.org/euclid.bams/1183510809available on-line from Project Euclid.)

Title	Sorgenfrey line
Canonical name	SorgenfreyLine
Date of creation	2013-03-22 13:03:45
Last modified on	2013-03-22 13:03:45
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	9
Author	yark (2760)
Entry type	Example
Classification	msc 55-00
Classification	msc 54-00
Classification	msc 22-00
Synonym	Sorgenfrey topology
Defines	lower limit topology