Editing Alexandroff extension (section)

{{Short description|Way to extend a non-compact topological space}}
In the [[mathematics|mathematical]] field of [[topology]], the '''Alexandroff extension''' is a way to extend a noncompact [[topological space]] by adjoining a single point in such a way that the resulting space is [[compact space|compact]].  It is named after the Russian mathematician [[Pavel Alexandroff]].
More precisely, let ''X'' be a topological space.  Then the Alexandroff extension of ''X'' is a certain compact space ''X''* together with an [[open mapping|open]] [[embedding (topology)|embedding]] ''c''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;''X''* such that the complement of ''X'' in ''X''* consists of a single point, typically denoted ∞.  The map ''c'' is a Hausdorff [[compactification (mathematics)|compactification]] if and only if ''X'' is a [[locally compact]], noncompact [[Hausdorff space]].  For such spaces the Alexandroff extension is called the '''one-point compactification''' or '''Alexandroff compactification'''.  The advantages of the Alexandroff compactification lie in its simple, often geometrically meaningful structure and the fact that it is in a precise sense minimal among all compactifications; the disadvantage lies in the fact that it only gives a Hausdorff compactification on the class of locally compact, noncompact Hausdorff spaces, unlike the [[Stone–Čech compactification]] which exists for any [[topological space]] (but [[Stone–Čech compactification#Universal property and functoriality|provides an embedding]] exactly for [[Tychonoff space|Tychonoff spaces]]).