Alexandrov one-point compactification

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

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

Title	Alexandrov one-point compactification
Canonical name	AlexandrovOnepointCompactification
Date of creation	2013-03-22 13:47:54
Last modified on	2013-03-22 13:47:54
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	9
Author	yark (2760)
Entry type	Definition
Classification	msc 54D35
Synonym	one-point compactification
Synonym	Alexandroff one-point compactification
Synonym	Aleksandrov one-point compactification
Synonym	Alexandrov compactification
Synonym	Aleksandrov compactification
Synonym	Alexandroff compactification
Related topic	Compactification