Editing Alexandroff extension (section)

== The Alexandroff extension ==
Let <math>X</math> be a topological space.  Put <math>X^* = X \cup \{\infty \},</math> and topologize <math>X^*</math> by taking as open sets all the open sets in ''X'' together with all sets of the form <math>V = (X \setminus C) \cup \{\infty \}</math> where ''C'' is closed and compact in ''X''.  Here, <math>X \setminus C</math> denotes the complement of <math> C</math> in <math>X.</math>  Note that <math>V</math> is an open neighborhood of <math>\infty,</math> and thus any open cover of  <math>\{\infty \}</math> will contain all except a compact subset <math>C</math> of <math>X^*,</math> implying that <math>X^*</math> is compact {{harv|Kelley|1975|p=150}}.

The space <math>X^*</math> is called the '''Alexandroff extension''' of ''X'' (Willard, 19A).  Sometimes the same name is used for the inclusion map <math>c: X\to X^*.</math>

The properties below follow from the above discussion:

* The map ''c'' is continuous and open: it embeds ''X'' as an open subset of <math>X^*</math>.
* The space <math>X^*</math> is compact.
* The image ''c''(''X'') is dense in <math>X^*</math>, if ''X'' is noncompact.
* The space <math>X^*</math> is [[Hausdorff space|Hausdorff]] if and only if ''X'' is Hausdorff and [[locally compact]].
* The space <math>X^*</math> is [[T1 space|T<sub>1</sub>]] if and only if ''X'' is T<sub>1</sub>.