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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.
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See Also: compactification

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

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something related to Alexandrov one-point compactification (Definition) by adrianita
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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 10293 times total.

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AMS MSC: 54D35 (General topology :: Fairly general properties :: Extensions of spaces )
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