nLab
end compactification

End compactification

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

End compactification

Idea

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

The theory of ends was invented by Hans Freudenthal in his dissertation, who gave a number of interesting applications (see below).

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

Consider the connected components of the complements XK iX \setminus K_i. An end of XX is an infinite sequence that chooses one such connected component for each ii. Remarkably, the set of ends is independent of the sequence KK chosen (up to natural bijection).

The end compactification of XX has, as its underlying set, the disjoint union of the underlying set of XX and the set of ends. Its topology is generated (from a base) by the topology of XX and, for each end e=(U 1,U 2,)e = (U_1,U_2,\ldots), the open sets V{e}V \cup \{e\} whenever VV is open in XX and U iVU_i \subseteq V for some (hence almost every) ii.

Abstract

Let XX be a topological space, and consider the poset Comp(X)Comp(X) of compact subspaces of XX, ordered by inclusion. For each compact subspace KK, consider its complement XKX \setminus K, and consider the set Π 0(XK)\Pi_0(X \setminus K) of its connected components. For each inclusion KKK \hookrightarrow K', we have a function Π 0(XK)Π 0(XK)\Pi_0(X \setminus K') \to \Pi_0(X \setminus K). This defines a contravariant functor from Comp(X)Comp(X) to Set; its limit is the set of ends of XX.

For the topology, each compact subspace KK defines a topological space KΠ 0(XK)K \uplus \Pi_0(X \setminus K); here, the points of Π 0(KK)\Pi_0(K \setminus K) are all isolated?. For each inclusion KKK \hookrightarrow K', we have a continuous map KΠ 0(XK)KΠ 0(XK)K' \uplus \Pi_0(X \setminus K') \to K \uplus \Pi_0(X \setminus K); it sends xx to itself if xKx \in K, and xx to the connected component [x]Π 0(XK)[x] \in \Pi_0(X \setminus K) if xKKx \in K' \setminus K. This defines a contravariant functor from Comp(X)Comp(X) to Top; its limit is the end compactification of XX.

Concrete

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

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, Freudenthal was led to his theory of ends by the following observation. For a space XX, consider a path-connected family FHomeo(X)F \subseteq Homeo(X) containing the identity 1 X1_X. Let KXK \subseteq X be compact, and let UU be a connected component of XKX \setminus K. Then for all fFf \in F, it may be shown f(U)Uf(U) \setminus U is contained in a compact subset of XX. The upshot is that ff extends to a homeomorphism on the end compactification that is pointwise fixed on the ends. If in addition for each pair (x,y)X 2(x, y) \in X^2 there is fFf \in F with f(x)=yf(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 3\mathbb{R}^3 cannot be given a topological group structure.

References

  • Wikipedia

  • G. Peschke, The Theory of Ends, Nieuw Archief voor Wiskunde, 8 (1990), 1–12. (pdf)

  • H. Freudenthal, Über die Enden topologischer Räume und Gruppen, Math. Z. 33 (1931), 692–713. (web)

Last revised on January 24, 2015 at 16:04:32. See the history of this page for a list of all contributions to it.