nLab
one-point compactification

Contents

Idea

The one-point compactification of a topological space is a new compact space obtained by adding a single new point to the original space.

This is also known as the Alexandroff compactification after a 1924 paper by Павел Сергеевич Александров (then transliterated ‘P.S. Aleksandroff’).

The one-point compactification is usually applied to a non-compact locally compact Hausdorff space. In the more general situation, it may not really be a compactification and hence is called the one-point extension or Alexandroff extension.

Definition

For topological spaces

Let XX be any topological space. Its one-point extension X *X^* is the topological space

(If XX is Hausdorff, then its compact subsets must always be closed, so (2) is often given in a simpler form.)

For non-commutative topological spaces (C *C^\ast-algebras)

Dually in non-commutative topology the one-point compactification corresponds to the unitisation of C*-algebras.

Properties

General

X *X^* is compact.

The evident inclusion XX *X \to X^* is an open embedding.

The one-point compactification is universal among all compact spaces into which XX has an open embedding, so it is essentially unique.

XX is dense in X *X^* iff XX is not already compact. Note that X *X^* is technically a compactification of XX only in this case.

X *X^* is Hausdorff (hence a compactum) if and only if XX is already both Hausdorff and locally compact.

Functoriality

The operation of one-point compactification is not a functor on the whole category of topological spaces. But it does extend to a functor on topological spaces with proper maps between them.

Examples

Spheres

For nn \in \mathbb{N} the nn-sphere (as a topological space) is the one-point compactification of the Cartesian space n\mathbb{R}^n

S n( n) *. S^n \simeq (\mathbb{R}^n)^\ast \,.

Via this presentation of the nn-sphere the canonical action of the orthogonal group O(N)O(N) on n\mathbb{R}^n induces an action of O(n)O(n) on S nS^n, which preserves the basepoint (the “point at infinity”).

This construction presents the J-homomorphism in stable homotopy theory and is encoded for instance in the definition of orthogonal spectra.

Slightly more generally, for VV any real vector space of dimension nn one has S n(V) *S^n \simeq (V)^\ast. In this context and in view of the previous case, one usually writes

S V(V) * S^V \coloneqq (V)^\ast

for the nn-sphere obtained as the one-point compactification of the vector space VV.

Proposition

For V,WVect V,W \in Vect_{\mathbb{R}} two real vector spaces, there is a natural homeomorphism

S VS WS VW S^V \wedge S^W \simeq S^{V\oplus W}

between the smash product of their one-point compactifications and the one-point compactification of the direct sum.

Remark

In particular, it follows directly from this that the suspension Σ()S 1()\Sigma(-) \simeq S^1 \wedge (-) of the nn-sphere is the (n+1)(n+1)-sphere, up to homeomorphism:

ΣS n S 1S n S 1 n S n+1 S n+1. \begin{aligned} \Sigma S^n & \simeq S^{\mathbb{R}^1} \wedge S^{\mathbb{R}^n} \\ & \simeq S^{\mathbb{R}^1 \oplus \mathbb{R}^n} \\ & \simeq S^{\mathbb{R}^{n+1}} \\ & \simeq S^{n+1} \end{aligned} \,.

Thom spaces

For XX a compact topological space and VXV \to X a vector bundle, then the (homotopy type of the) one-point compactification of the total space VV is the Thom space of VV, equivalent to D(V)/S(V)D(V)/S(V).

References

  • John Kelly, General Topology (1975)

Revised on November 11, 2013 07:59:37 by Urs Schreiber (89.204.139.93)