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'Sorgenfrey line'
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| Title of object: |
Sorgenfrey line |
| Canonical Name: |
SorgenfreyLine |
| Type: |
Example |
| Created on: |
2002-09-21 21:39:53 |
| Modified on: |
2004-11-08 23:14:13 |
| Classification: |
msc:54-00, msc:55-00, msc:22-00 |
| Defines: |
lower limit topology |
| Synonyms: |
Sorgenfrey line=Sorgenfrey topology |
Revision comment (for changes between this and next version):
Preamble:
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\def\sse{\subseteq}
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\def\R{\mathbb{R}} |
Content:
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
$\{1/n\}_n$ converges to $0$, while $\{-1/n\}_n$ does not.
This topology contains the standard topology on $\R$. The Sorgenfrey line is
first countable, separable, but not second countable. It is also not metrizable.
\begin{thebibliography}{9}
\bibitem{sorgenfrey} Sorgenfrey, R.~H. ``On the Topological Product of
Paracompact Spaces,'' Bulletin of the American Mathematical Society,
(1947) 631-632.
\end{thebibliography} |
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