# Alexandrov one-point compactification

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.

 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