compactify

Compactify is a term used in topology and related fields to describe the process of making a space compact by adjoining points or boundary.

A compactification of a topological space X is a pair (Y, i) where Y is compact and i: X → Y is a topological embedding with dense image. Equivalently, X is identified with a dense subspace of Y. The idea is to enlarge X to a compact space without changing its topology too much.

Two common compactifications are the one-point (Alexandroff) compactification and the Stone–Čech compactification. The one-point compactification applies to locally compact Hausdorff spaces X by adding a single point ∞; neighborhoods of ∞ correspond to complements of compact subsets of X. The resulting space Y = X ∪ {∞} is compact Hausdorff. The Stone–Čech compactification βX is the largest such compactification in a precise universal sense: βX is a compact Hausdorff space containing X densely such that any continuous map from X to any compact Hausdorff space extends uniquely to βX.

Other compactifications include Freudenthal end-compactifications for certain spaces. Compactifications are not unique; a given X can have many non-equivalent compactifications.

Examples help illustrate these ideas. The real line R has a one-point compactification homeomorphic to the circle S^1. The open interval (-1,1) compactifies to the closed interval [-1,1], obtained by adjoining endpoints. In algebraic geometry, compactification refers to embedding a variety into a complete (proper) variety, such as embedding affine space A^n into projective space P^n via projective completion.

Applications of compactification include extending functions, defining boundary conditions, and studying behavior at infinity in analysis, dynamics, and geometry.

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