Stone–Čech compactification

The Stone–Čech compactification βX of a completely regular space X is a compact Hausdorff space containing X densely, such that every continuous map from X into a compact Hausdorff space extends uniquely to βX.

Universal property: A property that characterizes an object by a unique mapping relationship to or from other objects. Every continuous map from X to a compact Hausdorff space extends uniquely through βX.
Completely regular space: A topological space in which points can be separated from closed sets by continuous real-valued functions. This is the exact class of spaces for which the Stone–Čech compactification is defined.
Stone–Čech compactification of the natural numbers: The compactification βℕ of the discrete natural numbers, whose points include free ultrafilters on ℕ. Its remainder contains the nonprincipal ultrafilters used in several areas of mathematics.
Alexandroff extension: The one-point compactification of a locally compact noncompact Hausdorff space. It adds one point, whereas βX is characterized by extending maps into every compact Hausdorff space.
Ultrafilter: A maximal proper filter on a set or Boolean algebra, representing a notion of convergence or consistent choice. Ultrafilters on X provide a concrete model for points of βX when X is discrete.
Compact space: A topological space in which every open cover has a finite subcover. Compactness is the defining global property of the space βX.
Čech–Stone remainder: The subspace βX \ X, consisting of the points added to X by its Stone–Čech compactification. The remainder isolates precisely what compactification adds beyond the original space.
Stone space: A compact Hausdorff totally disconnected space, equivalently the space associated with a Boolean algebra by Stone duality. βX need not be totally disconnected unless X’s topology imposes that structure.
Gelfand representation: A representation of a commutative unital C*-algebra as continuous functions on a compact Hausdorff space. The bounded continuous functions on X recover βX as their spectrum.
Hausdorff space: A topological space in which any two distinct points have disjoint neighborhoods. The compactification and its universal target spaces are required to be Hausdorff.

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