Editing Locally compact space (section)

=== The point at infinity ===
This section explores [[compactification (mathematics)|compactification]]s of locally compact spaces.  Every compact space is its own compactification.  So to avoid trivialities it is assumed below that the space ''X'' is not compact.

Since every locally compact Hausdorff space ''X'' is Tychonoff, it can be [[Embedding (topology)|embedded]] in a compact Hausdorff space <math>b(X)</math> using the [[Stone–Čech compactification]].
But in fact, there is a simpler method available in the locally compact case; the [[one-point compactification]] will embed ''X'' in a compact Hausdorff space <math>a(X)</math> with just one extra point.
(The one-point compactification can be applied to other spaces, but <math>a(X)</math> will be Hausdorff if and only if ''X'' is locally compact and Hausdorff.)
The locally compact Hausdorff spaces can thus be characterised as the open subsets of compact Hausdorff spaces.

Intuitively, the extra point in <math>a(X)</math> can be thought of as a '''point at infinity'''.
The point at infinity should be thought of as lying outside every compact subset of ''X''.
Many intuitive notions about tendency towards infinity can be formulated in locally compact Hausdorff spaces using this idea.
For example, a [[Continuous function (topology)|continuous]] [[real number|real]] or [[Complex number|complex]] valued [[Function (mathematics)|function]] ''f'' with [[Domain (function)|domain]] ''X'' is said to ''[[vanish at infinity]]'' if, given any [[positive number]] ''e'', there is a compact subset ''K'' of ''X'' such that <math>|f(x)| < e</math> whenever the [[Point (geometry)|point]] ''x'' lies outside of ''K''. This definition makes sense for any topological space ''X''. If ''X'' is locally compact and Hausdorff, such functions are precisely those extendable to a continuous function ''g'' on its one-point compactification <math>a(X) = X \cup \{ \infty \}</math> where <math>g(\infty) = 0.</math>