[[Compactification]]
# Alexandroff extension

The **Alexandroff extension** of a topological space $(X, \mathcal{T})$ is a [[Compact space]] $(X^\star, \mathcal{T}^\star)$ with an [[Open and closed maps|open]] [[Embedding|embedding]] $c : X \hookrightarrow X^\star$ such that $X^\star \setminus cX = \{ \omega \}$,
where $\omega$ is a special point often denoted $\infty$. #m/def/topology 
$$
\begin{align*}
U \in \mathcal{T}^\star \iff \begin{cases}
U \in \mathcal{T} & \omega \notin U \\
X  \setminus U \text{ compact}  & \omega \in U
\end{cases}
\end{align*}
$$

## Properties

- [[Proper map]]

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#state/tidy | #lang/en | #SemBr