The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

5.25 Stone-Čech compactification

The Stone-Čech compactification of a topological space $X$ is a map $X \to \beta (X)$ from $X$ to a Hausdorff quasi-compact space $\beta (X)$ which is universal for such maps. We prove this exists by a standard argument using the following simple lemma.

Lemma 5.25.1. Let $f : X \to Y$ be a continuous map of topological spaces. Assume that $f(X)$ is dense in $Y$ and that $Y$ is Hausdorff. Then the cardinality of $Y$ is at most the cardinality of $P(P(X))$ where $P$ is the power set operation.

Proof. Let $S = f(X) \subset Y$. Let $\mathcal{D}$ be the set of all closed domains of $Y$, i.e., subsets $D \subset Y$ which equal the closure of its interior. Note that the closure of an open subset of $Y$ is a closed domain. For $y \in Y$ consider the set

\[ I_ y = \{ T \subset S \mid \text{ there exists } D \in \mathcal{D}\text{ with }T = S \cap D\text{ and }y \in D\} . \]

Since $S$ is dense in $Y$ for every closed domain $D$ we see that $S \cap D$ is dense in $D$. Hence, if $D \cap S = D' \cap S$ for $D, D' \in \mathcal{D}$, then $D = D'$. Thus $I_ y = I_{y'}$ implies that $y = y'$ because the Hausdorff condition assures us that we can find a closed domain containing $y$ but not $y'$. The result follows. $\square$

Let $X$ be a topological space. By Lemma 5.25.1, there is a set $I$ of isomorphism classes of continuous maps $f : X \to Y$ which have dense image and where $Y$ is Hausdorff and quasi-compact. For $i \in I$ choose a representative $f_ i : X \to Y_ i$. Consider the map

\[ \prod f_ i : X \longrightarrow \prod \nolimits _{i \in I} Y_ i \]

and denote $\beta (X)$ the closure of the image. Since each $Y_ i$ is Hausdorff, so is $\beta (X)$. Since each $Y_ i$ is quasi-compact, so is $\beta (X)$ (use Theorem 5.14.4 and Lemma 5.12.3).

Let us show the canonical map $X \to \beta (X)$ satisfies the universal property with respect to maps to Hausdorff, quasi-compact spaces. Namely, let $f : X \to Y$ be such a morphism. Let $Z \subset Y$ be the closure of $f(X)$. Then $X \to Z$ is isomorphic to one of the maps $f_ i : X \to Y_ i$, say $f_{i_0} : X \to Y_{i_0}$. Thus $f$ factors as $X \to \beta (X) \to \prod Y_ i \to Y_{i_0} \cong Z \to Y$ as desired.

Lemma 5.25.2. Let $X$ be a Hausdorff, locally quasi-compact space. There exists a map $X \to X^*$ which identifies $X$ as an open subspace of a quasi-compact Hausdorff space $X^*$ such that $X^* \setminus X$ is a singleton (one point compactification). In particular, the map $X \to \beta (X)$ identifies $X$ with an open subspace of $\beta (X)$.

Proof. Set $X^* = X \amalg \{ \infty \} $. We declare a subset $V$ of $X^*$ to be open if either $V \subset X$ is open in $X$, or $\infty \in V$ and $U = V \cap X$ is an open of $X$ such that $X \setminus U$ is quasi-compact. We omit the verification that this defines a topology. It is clear that $X \to X^*$ identifies $X$ with an open subspace of $X$.

Since $X$ is locally quasi-compact, every point $x \in X$ has a quasi-compact neighbourhood $x \in E \subset X$. Then $E$ is closed (Lemma 5.12.3) and $V = (X \setminus E) \amalg \{ \infty \} $ is an open neighbourhood of $\infty $ disjoint from the interior of $E$. Thus $X^*$ is Hausdorff.

Let $X^* = \bigcup V_ i$ be an open covering. Then for some $i$, say $i_0$, we have $\infty \in V_{i_0}$. By construction $Z = X^* \setminus V_{i_0}$ is quasi-compact. Hence the covering $Z \subset \bigcup _{i \not= i_0} Z \cap V_ i$ has a finite refinement which implies that the given covering of $X^*$ has a finite refinement. Thus $X^*$ is quasi-compact.

The map $X \to X^*$ factors as $X \to \beta (X) \to X^*$ by the universal property of the Stone-Čech compactification. Let $\varphi : \beta (X) \to X^*$ be this factorization. Then $X \to \varphi ^{-1}(X)$ is a section to $\varphi ^{-1}(X) \to X$ hence has closed image (Lemma 5.3.3). Since the image of $X \to \beta (X)$ is dense we conclude that $X = \varphi ^{-1}(X)$. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0908. Beware of the difference between the letter 'O' and the digit '0'.



View Section 5.25 as pdf