Compactification

It is very useful to embed topological spaces in compact spaces, because of the strong properties compact spaces have. In general, the theory is limited to compactifications of Hausdorff spaces. For a locally compact space, there exists the one-point compatification: add a point at infinity, and define its neighborhoods as all co-compact sets. Because of local compactness, this topology is Haussdorf. Every Tychonoff space has the Stone-Cech compactifications, which is defined by the property that every continuous function to the reals can be continued to the compactification. Note that this does not hold for the one-point compactification: the one-point compactification of the Real line is a circle, but the function x |-> x cannot be continued continuously. Only Tychonoff spaces have Haussdorf compactifications, since a Hausdorff compact space is Tychonoff, and a subspace of a Tychonoff space is Tychnoff. This means that this is an alternative definition: a Tychonoff space is a Hausdorff space with compactification.