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^{}
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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 |