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\begin{document}

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\section*{compactification}

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

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

[[!include topology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{fiberwise}{Fiberwise}\dotfill \pageref*{fiberwise} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{uniqueness}{Uniqueness}\dotfill \pageref*{uniqueness} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{in_topology}{In topology}\dotfill \pageref*{in_topology} \linebreak
\noindent\hyperlink{in_geometry}{In geometry}\dotfill \pageref*{in_geometry} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Traditionally, a compactification of a [[topological space]] $X$ is a [[compact space]] $C$ together with an [[embedding]] $i \colon X \to C$ as a [[dense subspace]]. Well known is for instance the \emph{[[one-point compactification]]} of a [[locally compact Hausdorff space]] which for instance sends the real [[line]] to the [[circle]] by adding a point at infinity.

In some cases, the terminology `compactification' is applied to generalizations where $i$ is not a dense embedding (does not map [[homeomorphism|homeomorphically]] to an image that is dense). For example, the one-point compactification of an already [[compact Hausdorff space]] adds an \emph{isolated} point at infinity, and so is not a dense subspace. Or, if $X$ is not a Tychonoff space, then the universal map to its [[Stone-Cech compactification]] is not an embedding.

If the space has further geometric structure, the compactification is usually required to has such a structure and embedding has to preserve it. Many [[moduli space]]s in algebraic and differential geometry have their natural compactifications. They are often useful because they carry natural integration which is useful in defining various invariants.

A useful intuition throughout is that a `compactification' is a process of adding ``ideal points at infinity'' in some way to ``complete'' a space. (Compact [[regular space|regular]] spaces $X$ themselves being ``complete'' in a technical sense: there is a unique [[uniform space|uniform structure]] whose uniform topology is the topology on $X$, and $X$ is complete with respect to this uniformity.)

\hypertarget{fiberwise}{}\subsubsection*{{Fiberwise}}\label{fiberwise}

Often a space can be viewed as a total space of a bundle over some base. We may want to embed the space into a bigger bundle, such that the induced embedding of each fiber into the new fiber is a compactification.

This is roughly the case in most \textbf{compactifications in physics}. In most cases the space is equipped with a [[Riemannian metric]] and additional quantities for defining physics, like a [[Lagrangian density]], which possibly depend on the metric.

Then one requires that the compactified fiber is finite but small compared to some reference scale (or even viewed in a limit when the Riemannian volume tends to zero), see at \emph{[[Kaluza-Klein mechanism]]}. Often one does not even consider a noncompact case to start with but by compactification in physics means only passing to the limit of small (Riemannian volume of) fibers.

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

\begin{defn}
\label{Compactification}\hypertarget{Compactification}{}
A \emph{compactification} of a [[topological space]] $X$ is a [[compact topological space|compact]] [[Hausdorff topological space]] $Y$ equipped with an [[embedding]] $X \hookrightarrow Y$ such that the [[closure]] of $X$ in $Y$ is the compact space: $\overline{X} = Y$.

An [[equivalence]] of two compactifications $Y_1$, $Y_2$ of $X$ is a [[homeomorphism]] $h \;\colon\; Y_1 \longrightarrow Y_2$ that preserves the inclusion of $X$.

\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{uniqueness}{}\subsubsection*{{Uniqueness}}\label{uniqueness}

In some sense the [[one-point compactification]] is the smallest possible compactification, while the [[Stone-Cech compactification]] is the largest. The following gives conditions that all notions of compactification agree.

\begin{prop}
\label{UniqueCompactificationOfAlmostCompactSpaces}\hypertarget{UniqueCompactificationOfAlmostCompactSpaces}{}
For $X$ a [[Tychonoff space]] the following are equivalent:

\begin{enumerate}%
\item There is a unique (up to equivalence) compactification, def. \ref{Compactification}.


\item $X$ is already [[compact topological space|compact]] or its [[Stone-Cech compactification]] $\beta X$ adds a single point $\vert \beta X \backslash X\vert = 1$.


\item If two [[closed subsets]] of $X$ are completeley separated, then one of them is a [[compactum]].



\end{enumerate}
\end{prop}
One place where this appears is (\hyperlink{Hewitt47}{Hewitt 47}).

\begin{remark}
\label{}\hypertarget{}{}
The topological spaces satisfying the conditions of prop. \ref{UniqueCompactificationOfAlmostCompactSpaces} are also called [[almost compact topological space|almost compact topological spaces]].

\end{remark}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{in_topology}{}\subsubsection*{{In topology}}\label{in_topology}

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


\item [[Stone–Cech compactification]]


\item [[end compactification|End compactification]]


\item [[Bohr compactification]]


\item the [[Fulton-MacPherson operad]] is a compactification of [[configuration spaces of points]]



\end{itemize}
\hypertarget{in_geometry}{}\subsubsection*{{In geometry}}\label{in_geometry}

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


\item [[Deligne-Mumford compactification]].


\item [[Kaluza-Klein compactification]]

\begin{itemize}%
\item [[Freund-Rubin compactification]]

\end{itemize}

\item [[conformal compactification]]


\item [[projective compactification]]



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

\begin{itemize}%
\item Edwin Hewitt, \emph{Certain generalizations of the Weierstrass approximation theorem}, Duke Math. J. Volume 14, Number 2 (1947), 419-427. (\href{http://projecteuclid.org/euclid.dmj/1077474139}{Euclid})

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



\end{document}