nLab end compactification

Idea

Whereas the one-point compactification of a (sufficiently nice) topological space adjoins only a single point at infinity, the end compactification [Freudenthal 1931] adjoins one point for each connected component of infinity.

Definition

The definition was originally given only for sufficiently nice topological spaces (the hemicompact ones). The general definition is a bit more complicated. We will give three versions.

For hemicompact spaces only

Let $X$ be a topological space, and suppose that $X$ is hemicompact; this means that there exists an infinite sequence $n \mapsto K_n$ of compact subspaces of $X$ with $K_n \subseteq K_{n+1}$ such that every compact subspace of $X$ is contained in at least one (hence in almost all) of the $K_i$.

Consider the connected components of the complements $X \setminus K_i$. An end of $X$ is an infinite sequence that chooses one such connected component for each $i$. Remarkably, the set of ends is independent of the sequence $K$ chosen (up to natural bijection).

The end compactification of $X$ has, as its underlying set, the disjoint union of the underlying set of $X$ and the set of ends. Its topology is generated (from a base) by the topology of $X$ and, for each end $e = (U_1,U_2,\ldots)$, the open sets $V \cup \{e\}$ whenever $V$ is open in $X$ and $U_i \subseteq V$ for some (hence almost every) $i$.

Abstract

Let $X$ be a topological space, and consider the poset $Comp(X)$ of compact subspaces of $X$, ordered by inclusion. For each compact subspace $K$, consider its complement $X \setminus K$, and consider the set $\Pi_0(X \setminus K)$ of its connected components. For each inclusion $K \hookrightarrow K'$, we have a function $\Pi_0(X \setminus K') \to \Pi_0(X \setminus K)$. This defines a contravariant functor from $Comp(X)$ to Set; its limit is the set of ends of $X$.

For the topology, each compact subspace $K$ defines a topological space $K \uplus \Pi_0(X \setminus K)$; here, the points of $\Pi_0(K \setminus K)$ are all isolated. For each inclusion $K \hookrightarrow K'$, we have a continuous map $K' \uplus \Pi_0(X \setminus K') \to K \uplus \Pi_0(X \setminus K)$; it sends $x$ to itself if $x \in K$, and $x$ to the connected component $[x] \in \Pi_0(X \setminus K)$ if $x \in K' \setminus K$. This defines a contravariant functor from $Comp(X)$ to Top; its limit is the end compactification of $X$.

Concrete

Let $X$ be a topological space. An end of $X$ assigns, to each compact subspace $K$ of $X$, a connected component $e_K$ of its complement $X \setminus K$, in such a way that $e_{K'} \subseteq e_K$ whenever $K \subseteq K'$. The end compactification of $X$ has, as its underlying set, the disjoint union of the underlying set of $X$ and the set of ends. Its topology is generated (from a base) by the topology of $X$ and, for each end $e\colon K \mapsto e_K$, the open sets $V \cup \{e\}$ whenever $V$ is open in $X$ and $e_K \subseteq V$ for some compact subspace $K$.

Examples

A compact space has no ends, hence is its own end compactification. The converse (that a space with no ends must be compact) seems to require the axiom of choice (although excluded middle and dependent choice suffice for hemicompact spaces).

The end compactification of the real line is the extended real number line segment; the ends are $\infty$ and $-\infty$. But the complex plane has only one end; its end compactification is the Riemann sphere (the same as its one-point compactification).

Applications

Ends are important in proper homotopy theory.

According to Peschke 1990, Freudenthal 1931 was led to his theory of ends by the following observation. For a space $X$, consider a path-connected family $F \subseteq Homeo(X)$ containing the identity $1_X$. Let $K \subseteq X$ be compact, and let $U$ be a connected component of $X \setminus K$. Then for all $f \in F$, it may be shown $f(U) \setminus U$ is contained in a compact subset of $X$. The upshot is that $f$ extends to a homeomorphism on the end compactification that is pointwise fixed on the ends. If in addition for each pair $(x, y) \in X^2$ there is $f \in F$ with $f(x) = y$, then there is a severe constraint on the ends; in particular Freudenthal showed the following.

Theorem

A path-connected topological group has at most two ends.

For example, it follows that the space obtained by removing two points from $\mathbb{R}^3$ cannot be given a topological group structure.

Related concepts

• compactification

  • One-point compactification

  • Stone–Cech compactification

  • End compactification

  • Bohr compactification

References

The original text:

• Hans Freudenthal, Über die Enden topologischer Räume und Gruppen, Mathematische Zeitschrift 33 (1931) 692-713 [doi:10.1007/BF01174375, pdf]

Further discussion:

• Hans Freudenthal, Neuaufbau der Endentheorie, Annals of Mathematics Second Series 43 2 (1942) 261-279 [doi:10.2307/1968869, jstor:1968869]

• Hans Freudenthal, Über die Enden diskreter Räume und Gruppen, Commentarii Mathematici Helvetici 17 (1944) 1–38 [doi:10.1007/BF02566233]

• Georg Peschke, The Theory of Ends, Nieuw Archief voor Wiskunde, 8 (1990), 1-12 [pdf, pdf]

See also:

• Wikipedia, End (topology)