compactified

Compactified is the term used to describe a space or model that has undergone compactification, a process by which a non-compact space is enlarged to a compact one by adding boundary points or by changing its topology. The resulting space, called a compactification, contains the original space as a dense subset and satisfies a universal property with respect to maps into compact spaces.

Two common constructions are notable. The one-point (Alexandroff) compactification applies to any locally compact non-compact Hausdorff space X by adjoining a single point at infinity. The open neighborhoods of this point are complements of compact subsets of X, and the resulting space is compact and Hausdorff if X is. For example, the real line R becomes the circle S^1 under one-point compactification, and R^n becomes S^n.

The Stone–Čech compactification βX exists for any completely regular space X and is the largest compactification of X in a precise sense; X embeds densely into βX, and continuous bounded functions on X extend to βX.

In physics, compactification is used to describe extra spatial dimensions that are curled up at very small scales. Common shapes for the compact dimensions include circles, tori, and Calabi–Yau manifolds. Compactification yields a discrete spectrum of higher-dimensional fields (Kaluza-Klein modes) and can affect symmetries and the effective low-energy physics.