nLab
compactification

Contents

Idea

General

Traditionally, a compactification of a topological space XX is a compact space CC together with an embedding i:XCi \colon X \to C as a dense subspace. Well known is for instance the one-point compactification of a locally compact Hausdorff space which for instance sends the real line to the circle by adding a point at infinity.

In some cases, the terminology ‘compactification’ is applied to generalizations where ii is not a dense embedding (does not map homeomorphically to an image that is dense). For example, the one-point compactification of an already compact Hausdorff space adds an isolated point at infinity, and so is not a dense subspace. Or, if XX is not a Tychonoff space, then the universal map to its Stone-Cech compactification is not an embedding.

If the space has further geometric structure, the compactification is usually required to has such a structure and embedding has to preserve it. Many moduli spaces in algebraic and differential geometry have their natural compactifications. They are often useful because they carry natural integration which is useful in defining various invariants.

A useful intuition throughout is that a ‘compactification’ is a process of adding “ideal points at infinity” in some way to “complete” a space. (Compact regular spaces XX themselves being “complete” in a technical sense: there is a unique uniform structure whose uniform topology is the topology on XX, and XX is complete with respect to this uniformity.)

Fiberwise

Often a space can be viewed as a total space of a bundle over some base. We may want to embed the space into a bigger bundle, such that the induced embedding of each fiber into the new fiber is a compactification.

This is roughly the case in most compactifications in physics. In most cases the space is equipped with a Riemannian metric and additional quantities for defining physics, like a Lagrangian density, which possibly depend on the metric.

Then one requires that the compactified fiber is finite but small compared to some reference scale (or even viewed in a limit when the Riemannian volume tends to zero), see at Kaluza-Klein mechanism. Often one does not even consider a noncompact case to start with but by compactification in physics means only passing to the limit of small (Riemannian volume of) fibers.

Definition

Definition

A compactification of a topological space XX is a compact Hausdorff topological space YY equipped with an embedding XYX \hookrightarrow Y such that the closure of XX in YY is the compact space: X¯=Y\overline{X} = Y.

An equivalence of two compactifications Y 1Y_1, Y 2Y_2 of XX is a homeomorphism h:Y 1Y 2h \;\colon\; Y_1 \longrightarrow Y_2 that preserves the inclusion of XX.

Properties

Uniqueness

In some sense the one-point compactification is the smallest possible compactification, while the Stone-Cech compactification is the largest. The following gives conditions that all notions of compactification agree.

Proposition

For XX a Tychonoff space the following are equivalent:

  1. There is a unique (up to equivalence) compactification, def. 1.

  2. XX is already compact or its Stone-Cech compactification βX\beta X adds a single point βX\X=1\vert \beta X \backslash X\vert = 1.

  3. If two closed subsets of XX are completeley separated, then one of them is a compactum.

One place where this appears is (Hewitt 47).

Remark

The topological spaces satisfying the conditions of prop. 1 are also called almost compact topological spaces?.

Examples

In topology

In geometry

References

  • Edwin Hewitt, Certain generalizations of the Weierstrass approximation theorem, Duke Math. J. Volume 14, Number 2 (1947), 419-427. (Euclid)

Revised on December 3, 2013 13:20:26 by Toby Bartels (64.89.53.109)