Compactification

Compactification is a process in topology and geometry of enlarging a given space into a compact space in which the original space sits as a dense subspace. If Y is compact and X embeds densely into Y, Y is called a compactification of X (often denoted X̄). The choice of compactification encodes how one adds points at infinity to X.

One-point compactification: For a noncompact locally compact Hausdorff space X, there exists a compactification X̄ = X ∪ {∞} such that neighborhoods of ∞ are complements of compact subsets of X. If X is Hausdorff, X̄ is also Hausdorff. Examples include the real line R whose one-point compactification is homeomorphic to the circle S^1, and more generally R^n compactifying to S^n.

Stone–Čech compactification: For any completely regular (Tychonoff) space X, there is a universal compactification βX with an embedding X → βX such that every continuous map from X to a compact Hausdorff space K extends uniquely to a continuous map βX → K. βX is maximal among compactifications in a precise sense but can be hard to describe explicitly.

In algebraic geometry, compactification typically means a projective (proper) completion: an affine variety can be embedded into a projective variety by adding points at infinity; for example A^n embeds into P^n.

Applications: compactifications are used to study limits, boundary behavior of functions, and in physics where extra dimensions are hypothesized to be compactified on manifolds.