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 (210)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (37)
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
Alexandrov one-point compactification (Definition)

The Alexandrov one-point compactification of a non-compact topological space $X$ is obtained by adjoining a new point $\infty$ and defining the topology on $X\cup\{\infty\}$ to consist of the open sets of $X$ together with the sets of the form $U\cup\{\infty\}$, where $U$ is an open subset of $X$ with compact complement.

With this topology, $X\cup\{\infty\}$ is always compact. Furthermore, it is Hausdorff if and only if $X$ is Hausdorff and locally compact.



"Alexandrov one-point compactification" is owned by yark.
(view preamble)

View style:

See Also: compactification

Other names:  one-point compactification, Alexandroff one-point compactification, Aleksandrov one-point compactification, Alexandrov compactification, Aleksandrov compactification, Alexandroff compactification
Keywords:  compactification

Attachments:
something related to Alexandrov one-point compactification (Definition) by adrianita
Log in to rate this entry.
(view current ratings)

Cross-references: locally compact, Hausdorff, complement, compact, open sets, point, topological space
There are 9 references to this entry.

This is version 6 of Alexandrov one-point compactification, born on 2003-07-27, modified 2005-02-06.
Object id is 4515, canonical name is AlexandrovOnePointCompactification.
Accessed 9941 times total.

Classification:
AMS MSC: 54D35 (General topology :: Fairly general properties :: Extensions of spaces )
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)