Editing Compactification (mathematics) (section)

=== Alexandroff one-point compactification ===
{{main|One-point compactification}}
For any noncompact topological space ''X'' the ('''Alexandroff''') '''one-point compactification''' α''X'' of ''X'' is obtained by adding one extra point ∞ (often called a ''point at infinity'') and defining the [[open set]]s of the new space to be the open sets of ''X'' together with the sets of the form ''G''&nbsp;∪&nbsp;{{mset|∞}}, where ''G'' is an open subset of ''X'' such that <math>X \setminus G</math> is closed and compact. The one-point compactification of ''X'' is Hausdorff if and only if ''X'' is Hausdorff and [[locally compact]].<ref>{{citation|first=Pavel S.|last= Alexandroff|author-link=Pavel Alexandroff| journal=  [[Mathematische Annalen]] |volume= 92|issue=3–4 |year=1924|pages= 294–301|title= Über die Metrisation der im Kleinen kompakten topologischen Räume | url=https://eudml.org/doc/159072 | doi=10.1007/BF01448011 | jfm=50.0128.04 }}</ref>