nLab one-point compactification

Context

Topology

topology

algebraic topology

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

• whose underlying set is the disjoint union $X \cup \{\infty\}$

• and whose open sets are

1. the open subsets of $X$ (thought of as subsets of $X^*$);

2. the complements (in $X^*$) of the closed compact subsets of $X$.

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

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

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

Properties

General

$X^*$ is compact.

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

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

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

$X^*$ is Hausdorff (hence a compactum) if and only if $X$ 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 $n \in \mathbb{N}$ the $n$-sphere (as a topological space) is the one-point compactification of the Cartesian space $\mathbb{R}^n$

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

Via this presentation of the $n$-sphere the canonical action of the orthogonal group $O(N)$ on $\mathbb{R}^n$ induces an action of $O(n)$ on $S^n$, 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 $V$ any real vector space of dimension $n$ one has $S^n \simeq (V)^\ast$. In this context and in view of the previous case, one usually writes

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

for the $n$-sphere obtained as the one-point compactification of the vector space $V$.

Proposition

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

$S^V \wedge S^W \simeq S^{V\oplus W}$

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 $\Sigma(-) \simeq S^1 \wedge (-)$ of the $n$-sphere is the $(n+1)$-sphere, up to homeomorphism:

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

References

• John Kelly, General Topology (1975)

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