Use this tag for questions about making a topological space into a compact space.
In general topology, compactification is the process or result of making a topological space into a compact space. Formally, an embedding of a topological space X as a dense subset of a compact space is called a compactification of X.
It is often useful to embed topological spaces in compact spaces because they have special properties. Methods of compactification control points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape."
For any topological space X the (Alexandroff) one-point compactification αX of X is obtained by adding an extra point ∞ (often called a point at infinity) and defining the open sets of the new space to be the open sets of X together with the sets of the form G ∪ {∞} where G is an open subset of X such that X \ G is compact. The one-point compactification of X is Hausdorff if and only if X is Hausdorff and locally compact.
Of particular interest are Hausdorff compactifications, i.e., compactifications in which the compact space is Hausdorff. A topological space has a Hausdorff compactification if and only if it is Tychonoff. If so, there is a unique (up to homeomorphism) "most general" Hausdorff compactification, the Stone-Čech compactification of X, denoted by βX; formally, that exhibits the category of compact Hausdorff spaces and continuous maps as a reflective subcategory of the category of Tychonoff spaces and continuous maps.
"Most general," or, formally, "reflective," means that the space βX is characterized by the universal property that any continuous function from X to a compact Hausdorff space K can be extended uniquely to a continuous function from βX to K. More explicitly, βX is a compact Hausdorff space containing X such that the induced topology on X by βX is the same as the given topology on X, and for any continuous map f : X → K where K is a compact Hausdorff space, there is a unique continuous map g : βX → K for which g restricted to X is identically f.
The Stone-Čech compactification can be constructed as follows: let C be the set of continuous functions from X to the closed interval [0,1]. Then each point in X can be identified with an evaluation function on C. Thus X can be identified with a subset of [0,1]$^C$, the space of all functions from C to [0,1]. Because the latter is compact by Tychonoff's theorem, the closure of X as a subset of that space is also compact.
