PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (198)
Orphanage
Unclass'd
Unproven (490)
Corrections (7)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
[parent] Sorgenfrey line (Example)

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

$\displaystyle \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.

Bibliography

1
R. H. Sorgenfrey, On the topological product of paracompact spaces, Bulletin of the American Mathematical Society 53 (1947) 631-632. (This paper is available on-line from Project Euclid.)



"Sorgenfrey line" is owned by yark. [ full author list (2) | owner history (1) ]
(view preamble)

View style:

Other names:  Sorgenfrey topology
Also defines:  lower limit topology

This object's parent.

Attachments:
Sorgenfrey half-open plane (Definition) by rspuzio
Log in to rate this entry.
(view current ratings)

Cross-references: metrizable, second countable, separable, first countable, finer, points, limit from above, limit, standard topology, converges, sequence, open intervals, base, line, real, topology
There are 3 references to this entry.

This is version 6 of Sorgenfrey line, born on 2002-09-21, modified 2007-08-04.
Object id is 3469, canonical name is SorgenfreyLine.
Accessed 8225 times total.

Classification:
AMS MSC: 54-00 (General topology :: General reference works )
 55-00 (Algebraic topology :: General reference works )
 22-00 (Topological groups, Lie groups :: General reference works )
Pending Errata and Addenda
None.
[ View all 5 ]
Discussion
Style: Expand: Order:
forum policy
compact subsets of the Sorgenfrey line by bigli on 2006-12-09 04:21:28
Why any compact subset of sorgenfrey line must be a countable set?
[ reply | up ]
Is The Sorgenfrey topology a Baire space? by proximo on 2003-12-15 23:07:31
I know that the Sorgenfrey topology is totally disconnected, but I cant seem to prove that this implies it is a baire space. I have given up trying to prove directly that it is a baire space, I couldn't get anywhere with that.
[ reply | up ]
is the sorgenfrey line compact? by jancorod on 2003-05-09 11:49:00
Is the sorengfrey line compact?
[ reply | up ]
Interact
post | correct | update request | add example | add (any)