Editing Alexandroff extension (section)

== The one-point compactification ==
In particular, the Alexandroff extension <math>c: X \rightarrow X^*</math> is a Hausdorff compactification of ''X'' if and only if ''X'' is Hausdorff, noncompact and locally compact.  In this case it is called the '''one-point compactification''' or '''Alexandroff compactification''' of ''X''.

Recall from the above discussion that any Hausdorff compactification with one point remainder is necessarily (isomorphic to) the Alexandroff compactification.  In particular, if <math>X</math> is a compact Hausdorff space and <math>p</math> is a [[limit point]] of <math>X</math> (i.e. not an [[isolated point]] of <math>X</math>), <math>X</math> is the Alexandroff compactification of <math>X\setminus\{p\}</math>.

Let ''X'' be any noncompact [[Tychonoff space]].  Under the natural partial ordering on the set <math>\mathcal{C}(X)</math> of equivalence classes of compactifications, any minimal element is equivalent to the Alexandroff extension (Engelking, Theorem 3.5.12).  It follows that a noncompact Tychonoff space admits a minimal compactification if and only if it is locally compact.