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\section*{end compactification}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{end_compactification}{}\section*{{End compactification}}\label{end_compactification}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{for_hemicompact_spaces_only}{For hemicompact spaces only}\dotfill \pageref*{for_hemicompact_spaces_only} \linebreak
\noindent\hyperlink{abstract}{Abstract}\dotfill \pageref*{abstract} \linebreak
\noindent\hyperlink{concrete}{Concrete}\dotfill \pageref*{concrete} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Whereas the [[one-point compactification]] of a (sufficiently [[nice topological space|nice]]) [[topological space]] adjoins only a single [[point at infinity]], the \emph{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).

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

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

\hypertarget{for_hemicompact_spaces_only}{}\subsubsection*{{For hemicompact spaces only}}\label{for_hemicompact_spaces_only}

Let $X$ be a [[topological space]], and suppose that $X$ is [[hemicompact space|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 \textbf{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 \textbf{end compactification} of $X$ has, as its [[underlying set]], the [[disjoint union]] of the underlying set of $X$ and the set of ends. Its [[topological structure|topology]] is generated (from a [[topological base|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) $i$.

\hypertarget{abstract}{}\subsubsection*{{Abstract}}\label{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 \textbf{set of ends} of $X$.

For the [[topological structure|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 point|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 \textbf{end compactification} of $X$.

\hypertarget{concrete}{}\subsubsection*{{Concrete}}\label{concrete}

Let $X$ be a [[topological space]]. An \textbf{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 \textbf{end compactification} of $X$ has, as its [[underlying set]], the [[disjoint union]] of the underlying set of $X$ and the set of ends. Its [[topological structure|topology]] is generated (from a [[topological base|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 $K$.

\hypertarget{examples}{}\subsection*{{Examples}}\label{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]]).

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

Ends are important in [[proper homotopy theory]].

According to \hyperlink{Peschke}{Peschke}, Freudenthal 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 \emph{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.

\begin{theorem}
\label{}\hypertarget{}{}
A path-connected topological group has at most two ends.

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

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[compactification]]

\begin{itemize}%
\item [[One-point compactification]]


\item [[Stone–Cech compactification]]


\item \textbf{End compactification}


\item [[Bohr compactification]]



\end{itemize}


\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item \href{https://en.wikipedia.org/wiki/End_%28topology%29}{Wikipedia}


\item G. Peschke, \emph{The Theory of Ends}, Nieuw Archief voor Wiskunde, 8 (1990), 1--12. (\href{https://www.ualberta.ca/~gepe/pdf/Peschke_TheoryOfEnds.pdf}{pdf})



\end{itemize}
\begin{itemize}%
\item H. Freudenthal, \emph{\"U{}ber die Enden topologischer R\"a{}ume und Gruppen}, Math. Z. 33 (1931), 692--713. (\href{http://dspace.library.uu.nl/bitstream/handle/1874/7437/1930-freudenthal-dissertatie.pdf?sequence=1}{web})

\end{itemize}
[[!redirects end compactification]] [[!redirects end compactifications]]

[[!redirects topological end]] [[!redirects topological ends]] [[!redirects end of a topological space]] [[!redirects ends of a topological space]] [[!redirects ends of topological spaces]]



\end{document}