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\section*{one-point 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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{for_topological_spaces}{For topological spaces}\dotfill \pageref*{for_topological_spaces} \linebreak
\noindent\hyperlink{for_noncommutative_topological_spaces_algebras}{For non-commutative topological spaces ($C^\ast$-algebras)}\dotfill \pageref*{for_noncommutative_topological_spaces_algebras} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{BasicProperties}{Basic properties}\dotfill \pageref*{BasicProperties} \linebreak
\noindent\hyperlink{UniversalProperty}{Universal property}\dotfill \pageref*{UniversalProperty} \linebreak
\noindent\hyperlink{functoriality}{Functoriality}\dotfill \pageref*{functoriality} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{ExamplesSpheres}{Eculidean spaces compactify to Spheres}\dotfill \pageref*{ExamplesSpheres} \linebreak
\noindent\hyperlink{linear_representations_compactify_to_representation_spheres}{Linear representations compactify to representation spheres}\dotfill \pageref*{linear_representations_compactify_to_representation_spheres} \linebreak
\noindent\hyperlink{thom_spaces}{Thom spaces}\dotfill \pageref*{thom_spaces} \linebreak
\noindent\hyperlink{locally_compact_hausdorff_spaces}{Locally compact Hausdorff spaces}\dotfill \pageref*{locally_compact_hausdorff_spaces} \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}

The \emph{one-point compactification} of a [[topological space]] $X$ is a new [[compact space]] $X^* = X \cup \{\infty\}$ obtained by adding a single new point ``$\infty$'' to the original space and declaring in $X^*$ the [[complements]] of the original [[closed subspace|closed]] [[compact space|compact]] [[subspaces]] to be [[open subspace|open]].

One may think of the new point added as the ``point at infinity'' of the original space. A [[continuous function]] on $X$ \emph{[[vanishing at infinity|vanishes at infinity]]} precisely if it extends to a continuous function on $X^*$ and literally takes the value zero at the point ``$\infty$''.

This one-point compactification is also known as the \emph{Alexandroff compactification} after a 1924 paper by [[Pavel Aleksandrov|Павел Сергеевич Александров]] (then transliterated `P.S. Aleksandroff').

The one-point compactification is usually applied to a non-[[compact space|compact]] [[locally compact space|locally compact]] [[Hausdorff space|Hausdorff]] space. In the more general situation, it may not really be a [[compactification]] and hence is called the \emph{one-point extension} or \emph{Alexandroff extension}.

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

\hypertarget{for_topological_spaces}{}\subsubsection*{{For topological spaces}}\label{for_topological_spaces}

\begin{defn}
\label{OnePointExtension}\hypertarget{OnePointExtension}{}
\textbf{(one-point extension)}

Let $X$ be any [[topological space]]. Its \textbf{one-point extension} $X^*$ is the topological space

\begin{itemize}%
\item whose underlying [[set]] is the [[disjoint union]] $X \cup \{\infty\}$


\item and whose [[open sets]] are

\begin{enumerate}%
\item the open subsets of $X$ (thought of as subsets of $X^*$);


\item the [[complements]] $X^\ast \backslash CK = (X \backslash CK) \cup \{\infty\}$ of the [[closed subspace|closed]] [[compact space|compact]] subsets $CK \subset X$.



\end{enumerate}


\end{itemize}
\end{defn}
(\hyperlink{Aleksandrov24}{Aleksandrov 24}, see \hyperlink{Kelly75}{Kelly 75, p. 150})

\begin{remark}
\label{}\hypertarget{}{}
If $X$ is [[Hausdorff space|Hausdorff]], then it is sufficient to speak of compact subsets in def. \ref{OnePointExtension}, since [[compact subspaces of Hausdorff spaces are closed]].

\end{remark}
\begin{lemma}
\label{OnePointExtensionWellDefined}\hypertarget{OnePointExtensionWellDefined}{}
\textbf{(one-point extension is well-defined)}

The [[topological space|topology]] on the one-point extension in def. \ref{OnePointExtension} is indeed well defined in that the given set of subsets is indeed closed under arbitrary unions and finite intersections.

\end{lemma}
\begin{proof}
The unions and finite intersections of the open subsets inherited from $X$ are closed among themselves by the assumption that $X$ is a topological space.

It is hence sufficient to see that

\begin{enumerate}%
\item the unions and finite intersection of the $(X \backslash CK) \cup \{\infty\}$ are closed among themselves,


\item the union and intersection of a subset of the form $U \underset{\text{open}}{\subset} X \subset X^\ast$ with one of the form $(X \backslash CK) \cup \{\infty\}$ is again of one of the two kinds.



\end{enumerate}
Regarding the first statement: Under [[de Morgan duality]]

\begin{displaymath}
\underset{i \in \underset{\text{finite}}{J}}{\bigcap} (X \backslash CK_i \cup \{\infty\})
  = 
  \left( X \backslash \left(\underset{i \in \underset{\text{finite}}{J}}{\bigcup} CK_i \right)\right) \cup \{\infty\}
\end{displaymath}
and

\begin{displaymath}
\underset{i \in I}{\bigcup} ( X \backslash CK_i \cup \{\infty\} )
  = 
  \left(X \backslash \left(\underset{i \in I}{\bigcap} CK_i \right)\right) \cup \{\infty\}
\end{displaymath}
and so the first statement follows from the fact that finite unions of compact subspaces and arbitrary intersections of closed compact subspaces are themselves again compact (\href{compact+space#UnionsAndIntersectionOfCompactSubspaces}{this prop.}).

Regarding the second statement: That $U \subset X$ is open means that there exists a closed subset $C \subset X$ with $U = X\backslash C$. Now using [[de Morgan duality]] we find

\begin{enumerate}%
\item for intersections:

\begin{displaymath}
\begin{aligned}
    U \cap ( (X\backslash CK) \cup \{\infty\} )
    & = 
    (X \backslash C) \cap (X \backslash CK)
    \\
    & =
    X \backslash (C \cup CK).
  \end{aligned}
\end{displaymath}
Since finite unions of closed subsets are closed, this is again an open subset of $X$;


\item for unions:

\begin{displaymath}
\begin{aligned}
    U \cup (X \backslash CK) \cup \{\infty\}
    & = 
    (X \backslash C) \cup (X \backslash CK) \cup \{\infty\}
    \\
    & =
    (X \backslash (C \cap CK)) \cup \{\infty\} .
  \end{aligned}
\end{displaymath}
For this to be open in $X^\ast$ we need that $C \cap CK$ is again compact. This follows because [[subsets are closed in a closed subspace precisely if they are closed in the ambient space]] and because [[closed subsets of compact spaces are compact]].



\end{enumerate}
\end{proof}
\hypertarget{for_noncommutative_topological_spaces_algebras}{}\subsubsection*{{For non-commutative topological spaces ($C^\ast$-algebras)}}\label{for_noncommutative_topological_spaces_algebras}

Dually in [[non-commutative topology]] the one-point compactification corresponds to the [[unitisation of C\emph{-algebras]].}

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

\hypertarget{BasicProperties}{}\subsubsection*{{Basic properties}}\label{BasicProperties}

We discuss the basic properties of the construction $X^\ast$ in def. \ref{OnePointExtension}, in particular that it always yields a [[compact topological space]] (prop. \ref{OnePointExtensionIsCompact} below) and the ingredients needed to see its [[universal property]] in the Hausdorff case \hyperlink{OnePointExtensionIsCompact}{below}.

\begin{prop}
\label{OnePointExtensionIsCompact}\hypertarget{OnePointExtensionIsCompact}{}
\textbf{(one-point extension is compact)}

For $X$ any [[topological space]], we have that its one-point extension $X^\ast$ (def. \ref{OnePointExtension}) is a [[compact topological space]].

\end{prop}
\begin{proof}
Let $\{U_i \subset X^\ast\}_{i \in I}$ be an [[open cover]]. We need to show that this has a finite subcover.

That we have a cover means that

\begin{enumerate}%
\item there must exist $i_\infty \in I$ such that $U_{i \infty} \supset \{\infty\}$ is an [[open neighbourhood]] of the extra point. But since, by construction, the only open subsets containing that point are of the form $(X \backslash CK) \cup \{\infty\}$, it follows that there is a compact closed subset $CK \subset X$ with $X \backslash CK \subset U_{i \infty}$.


\item $\{U_i \subset X\}_{i \in i}$ is in particular an open cover of that closed compact subset $CK \subset X$. This being compact means that there is a finite subset $J \subset I$ so that $\{U_i \subset X\}_{i \in \J \subset X}$ is still a cover of $CK$.



\end{enumerate}
Together this implies that

\begin{displaymath}
\{U_i \subset X\}_{i \in J \subset I}
  \cup
  \{ U_{i_\infty} \}
\end{displaymath}
is a finite subcover of the original cover.

\end{proof}
\begin{prop}
\label{HausdorffOnePointCompactification}\hypertarget{HausdorffOnePointCompactification}{}
\textbf{(one-point extension of locally compact space is Hausdorff precisely if original space is)}

Let $X$ be a [[locally compact topological space]]. Then its one-point extension $X^\ast$ (def. \ref{OnePointExtension}) is a [[Hausdorff topological space]] precisely if $X$ is.

\end{prop}
\begin{proof}
It is clear that if $X$ is not Hausdorff then $X^\ast$ is not.

For the converse, assume that $X$ is Hausdorff.

Since $X^\ast = X \cup \{\infty\}$ as underlying sets, we only need to check that for $x \in X$ any point, then there is an open neighbourhood $U_x \subset X \subset X^\ast$ and an open neighbourhood $V_\infty \subset X^\ast$ of the extra point which are disjoint.

That $X$ is locally compact implies by definition that there exists an open neighbourhood $U_k \supset \{x\}$ whose [[topological closure]] $CK \coloneqq Cl(U_x)$ is a closed compact neighbourhood $CK \supset \{x\}$. Hence

\begin{displaymath}
V_\infty \coloneqq (X \backslash CK ) \cup \{\infty\} \subset X^\ast
\end{displaymath}
is an open neighbourhood of $\{\infty\}$ and the two are disjoint

\begin{displaymath}
U_x \cap V_\infty = \emptyset
\end{displaymath}
by construction.

\end{proof}
\begin{prop}
\label{InclusionIntoOnePointExtensionIsOpenEmbedding}\hypertarget{InclusionIntoOnePointExtensionIsOpenEmbedding}{}
\textbf{(inclusion into one-point extension is open embedding)}

Let $X$ be a [[topological space]]. Then the evident inclusion function

\begin{displaymath}
i \;\colon\; X \longrightarrow X^\ast
\end{displaymath}
into its one-point extension (def. \ref{OnePointExtension}) is

\begin{enumerate}%
\item a [[continuous function]]


\item an [[open map]]


\item an [[embedding of topological spaces]].



\end{enumerate}
\end{prop}
\begin{proof}
Regarding the first point: For $U \subset X$ open and $CK \subset X$ closed and compact, the preimages of the corresponding open subsets in $X^\ast$ are

\begin{displaymath}
i^{-1}(U) = U
  \phantom{AAAA}
  i^{-1}( (X \backslash CK) \cup \infty ) = X \backslash CK
\end{displaymath}
which are open in $X$.

Regarding the second point: The image of an open subset $U \subset X$ is $i(U) = U \subset X^\ast$, which is open by definition.

Regarding the third point: We need to show that $i \colon X \to i(X) \subset X^\ast$ is a [[homeomorphism]]. This is immediate from the definition of $X^\ast$.

\end{proof}
\begin{remark}
\label{CompactHausdorffSpaceIsCompactificationOfComplementOfAnyPoint}\hypertarget{CompactHausdorffSpaceIsCompactificationOfComplementOfAnyPoint}{}
For $X$ a [[compact Hausdorff space]] and $x_0 \in X$ any point, then $X$ is [[homeomorphism|homeomorphic]] to the one-point compactification of the [[complement]] [[subspace]] $X \setminus \{x_0\} \subset X$:

\begin{displaymath}
X \simeq (X \setminus \{x_0\})^\ast
  \,.
\end{displaymath}
Observe also that $X \setminus \{x_0\}$, being an [[open subspace]] of a [[compact Hausdorff space]] is a [[locally compact topological space]], since [[open subspaces of compact Hausdorff spaces are locally compact]], and of course it is Hausdorff, since $X$ is.

\end{remark}
\begin{proof}
Since [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]], the open neighbourhoods of $x \in X$ are equivalently the complements of closed, and hence compact closed, subsets in $X \setminus \{x\}$. By def. \ref{OnePointExtension} this means that the function

\begin{displaymath}
\itexarray{
     X &\longrightarrow& (X \setminus \{x_0\})^\ast
  }
\end{displaymath}
which is the identity on $X \setminus \{x_0\}$ and sends $x_0 \mapsto \infty$ (hence which is just the identity on the underlying sets) is a [[homeomorphism]].

\end{proof}
\hypertarget{UniversalProperty}{}\subsubsection*{{Universal property}}\label{UniversalProperty}

As a [[pointed topological space|pointed]] [[locally compact topological space|locally compact]] [[Hausdorff space]], the one-point compactification of $X$ may be described by a [[universal property]]:

For every [[pointed topological space|pointed]] [[locally compact topological space|locally compact]] [[Hausdorff space]] $(Y, y_0)$ and every [[continuous map]] $f \colon X \to Y$ such that the [[pre-image]] $f^{-1}(K)$ is compact for all compact sets $K$ not containing $y_0$, there is a unique basepoint-preserving continuous map $X^\ast \to Y$ that extends $f$.

This property characterizes $X^\ast$ in an [[essentially unique]] manner.

$X$ is [[dense subspace|dense]] in $X^*$ precisely if $X$ is not already compact. Note that $X^*$ is technically a [[compactification]] of $X$ only in this case.

$X^*$ is [[Hausdorff space|Hausdorff]] (hence a [[compactum]]) if and only if $X$ is already both Hausdorff and [[locally compact space|locally compact]] (see prop. \ref{HausdorffOnePointCompactification}).

\hypertarget{functoriality}{}\subsubsection*{{Functoriality}}\label{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.

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

\hypertarget{ExamplesSpheres}{}\subsubsection*{{Eculidean spaces compactify to Spheres}}\label{ExamplesSpheres}

We discuss how the [[one-point compactification]] of [[Euclidean space]] of [[dimension]] $n$ is the [[n-sphere]].

\begin{example}
\label{nSphereIsOnePointCompactificationOfRn}\hypertarget{nSphereIsOnePointCompactificationOfRn}{}
\textbf{([[one-point compactification]] of [[Euclidean space|Euclidean n-space]] si the [[n-sphere]])}

For $n \in \mathbb{N}$ the [[n-sphere]] with its standard [[topological space|topology]] (e.g. as a [[subspace]] of the [[Euclidean space]] $\mathbb{R}^{n+1}$ with its [[metric topology]]) is [[homeomorphism|homeomorphic]] to the one-point compactification (def. \ref{OnePointExtension}) of the [[Euclidean space]] $\mathbb{R}^n$

\begin{displaymath}
S^n \simeq (\mathbb{R}^n)^\ast
  \,.
\end{displaymath}
\end{example}
\begin{proof}
Pick a point $\infty \in S^n$. By [[stereographic projection]] we have a [[homeomorphism]]

\begin{displaymath}
S^n \setminus \{\infty\} \simeq \mathbb{R}^n
  \,.
\end{displaymath}
With this it only remains to see that for $U_\infty \supset \{\infty\}$ an open neighbourhood of $\infty$ in $S^n$ then the complement $S^n \setminus U_\infty$ is compact closed, and cnversely that the complement of every compact closed subset of $S^n \setminus \{\infty\}$ is an open neighbourhood of $\{\infty\}$.

Observe that under [[stereographic projection]] the open subspaces $U_\infty \setminus \{\infty\} \subset S^n \setminus \{\infty\}$ are identified precisely with the [[closed subset|closed]] and [[bounded subsets]] of $\mathbb{R}^n$. (Closure is immediate, boundedness follows because an open neighbourhood of $\{\infty\} \in S^n$ needs to contain an open ball around $0 \in \mathbb{R}^n \simeq S^n \setminus \{-\infty\}$ in the \emph{other} stereographic projection, which under change of chart gives a bounded subset. )

By the [[Heine-Borel theorem]] the closed and bounded subsets of $\mathbb{R}^n$ are precisely the compact, and hence the compact closed, subsets of $\mathbb{R}^n \simeq S^n \setminus \{\infty\}$.

\end{proof}
\begin{remark}
\label{RelevanceForMonopolesAndInstantons}\hypertarget{RelevanceForMonopolesAndInstantons}{}
\textbf{(relevance for [[monopoles]] and [[instantons]] in [[gauge theory]])}

In [[physics]], Example \ref{nSphereIsOnePointCompactificationOfRn} governs the phenomenon of [[monopoles]] and [[instantons]] for [[gauge theory]] on [[Minkowski spacetime]] or [[Euclidean space]]: While such spaces themselves are not [[compact topological space|compact]], the consistency condition that any [[field history|field configuration]] carries a [[finite number|finite]] [[energy]] requires that [[gauge fields]] \emph{[[vanishing at infinity|vanish at infinity]]}.

This means that if $A$ is the [[classifying space]] for the corresponding gauge field -- e.g. $A = B G$ for [[Yang-Mills theory]] with [[gauge group]] $G$ -- and if $a_\infty \in A$ denotes the [[pointed topological space|base point]] witnessing vanishing fields, then a [[field configuration]]/[[cocycle]]

\begin{displaymath}
\mathbb{R}^n
  \overset{c}{\longrightarrow}
  A
\end{displaymath}
on $\mathbb{R}^n$ in $A$-[[generalized cohomology theory|cohomology]] \emph{[[vanishing at infinity|vanishes at infinity]]} if outside any [[compact topological space|compact]] [[subset]] its value is the vanishing field configuration $a_\infty$.

But by Def. \ref{OnePointExtension} this is equivalent to the cocycle [[extension|extends]] to the [[one-point compactification]] as a [[morphism]] of [[pointed topological space]]:

\begin{displaymath}
\itexarray{
    \big(
      \mathbb{R}^n
    \big)
    \simeq S^n
    &
    \overset{c}{\longrightarrow}
    &
    A
    \\
    \infty &\mapsto& a_\infty
  }
\end{displaymath}
The following graphics illustrates this for $A = S^n$ an [[n-sphere]] itself, hence for charges in [[Cohomotopy cohomology theory]]:

\begin{quote}%
graphics grabbed from \href{cohomotopy+charge#SatiSchreiber19}{SS 19}


\end{quote}
For more see at \emph{\href{Yang-Mills+instanton#FromTheMathsToThePhysicsStory}{Yang-Mills instanton -- SU(2)-instantons from the correct maths to the traditional physics story}}.

\end{remark}
\hypertarget{linear_representations_compactify_to_representation_spheres}{}\subsubsection*{{Linear representations compactify to representation spheres}}\label{linear_representations_compactify_to_representation_spheres}

Via the presentation of example \ref{nSphereIsOnePointCompactificationOfRn}, 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 $\infty$ (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

\begin{displaymath}
S^V \coloneqq (V)^\ast
\end{displaymath}
for the $n$-[[sphere]] obtained as the one-point compactification of the vector space $V$.

\begin{prop}
\label{}\hypertarget{}{}
For $V,W \in Vect_{\mathbb{R}}$ two real [[vector spaces]], there is a [[natural transformation|natural]] [[homeomorphism]]

\begin{displaymath}
S^V \wedge S^W \simeq S^{V\oplus W}
\end{displaymath}
between the [[smash product]] of their one-point compactifications and the one-point compactification of the [[direct sum]].

\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
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{displaymath}
\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}
  \,.
\end{displaymath}
\end{remark}
$\backslash$linebreak

\hypertarget{thom_spaces}{}\subsubsection*{{Thom spaces}}\label{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 $D(V)/S(V)$.

For a simple example: the real [[projective plane]] $\mathbb{RP}^2$ is the one-point compactification of the `open' [[Moebius strip|Möbius strip]], as line bundle over $S^1$. This is a special case of the more general observation that $\mathbb{RP}^{n+1}$ is the Thom space of the [[tautological line bundle]] over $\mathbb{RP}^n$.

\hypertarget{locally_compact_hausdorff_spaces}{}\subsubsection*{{Locally compact Hausdorff spaces}}\label{locally_compact_hausdorff_spaces}

\begin{example}
\label{LocallyCompatcHausdorffSpaceIsOpenSubspaceOfCompactHausdorffSpace}\hypertarget{LocallyCompatcHausdorffSpaceIsOpenSubspaceOfCompactHausdorffSpace}{}
\textbf{(every [[locally compact topological space|locally compact]] [[Hausdorff space]] is an [[open subset|open]] [[subspace]] of a [[compact Hausdorff space]])}

Every [[locally compact topological space|locally compact]] [[Hausdorff space]] is [[homeomorphism|homemorphic]] to a [[open subset|open]] [[topological subspace]] of a [[compact topological space]].

\end{example}
\begin{proof}
In one direction the statement is that [[open subspaces of compact Hausdorff spaces are locally compact]] (see there for the proof). What we need to show is that every locally compact Hausdorff spaces arises this way.

So let $X$ be a locally compact Hausdorff space. By prop. \ref{OnePointExtensionIsCompact} and prop. \ref{HausdorffOnePointCompactification} its one-point extension $X^\ast$ (def. \ref{OnePointExtension}) is a [[compact Hausdorff space]]. By prop. \ref{InclusionIntoOnePointExtensionIsOpenEmbedding} the canonical inclusion $X \to X^\ast$ is an [[open map|open]] [[embedding of topological spaces]].

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

\begin{itemize}%
\item [[Thom space]]


\item [[compactification]]

\begin{itemize}%
\item [[Stone-Cech compactification]]


\item [[end compactification]]


\item [[Bohr compactification]]


\item [[Kaluza-Klein compactification]]



\end{itemize}

\item [[configuration space (mathematics)]]



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

The concept goes back to

\begin{itemize}%
\item [[Pavel Aleksandrov]], \emph{Über die Metrisation der im Kleinen kompakten topologischen Räume}, Mathematische Annalen (1924) Volume: 92, page 294-301 (\href{https://eudml.org/doc/159072}{dml:159072})

\end{itemize}
Textbook accounts:

\begin{itemize}%
\item John Kelly, \href{https://archive.org/details/GeneralTopology/page/n167}{p. 150} of: \emph{General Topology}, van Nostrand 1955 (\href{https://archive.org/details/GeneralTopology}{archive:GeneralTopology})

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Alexandroff_extension}{Alexandroff extension}}

\end{itemize}
[[!redirects one-point compactification]] [[!redirects one-point compactifications]] [[!redirects One-point compactification]] [[!redirects Alexandroff compactification]] [[!redirects Alexandroff compactifications]] [[!redirects Alexandrov compactification]] [[!redirects Alexandrov compactifications]] [[!redirects Aleksandrov compactification]] [[!redirects Aleksandrov compactifications]] [[!redirects Александров compactification]] [[!redirects Александров compactifications]]

[[!redirects one-point extension]] [[!redirects one-point extensions]] [[!redirects Alexandroff extension]] [[!redirects Alexandroff extensions]] [[!redirects Alexandrov extension]] [[!redirects Alexandrov extensions]] [[!redirects Aleksandrov extension]] [[!redirects Aleksandrov extensions]] [[!redirects Александров extension]] [[!redirects Александров extensions]]

[[!redirects point at infinity]]



\end{document}