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Alexandrov one-point compactification
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(Definition)
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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.
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"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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Cross-references: locally compact, Hausdorff, complement, compact, open sets, point, topological space
There are 10 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 13453 times total.
Classification:
| AMS MSC: | 54D35 (General topology :: Fairly general properties :: Extensions of spaces ) |
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Pending Errata and Addenda
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