Definition 5.25.2 (Stone-Čech Compactification).label Let $X$ be a completely regular space, then there exists a pair $(\beta X, e)$ such that:
- (1)
$(\beta X, e)$ is a compactification of $X$.
- (U1)
For any $f \in C(X; [0, 1])$, there exists a unique $\beta f \in C(\beta X; [0, 1])$ such that the following diagram commutes:
\[\xymatrix{ \beta X \ar@{->}[r]^{\beta f} & [0, 1] \\ X \ar@{->}[u]^{e} \ar@{->}[ru]_{f} & }\]
Moreover, if $(\beta X, e)$ is any pair that satisfies (1) and (U1), then
- (U2)
For any pair $(Y, \varphi)$ satisfying (1), there exists a unique $\beta \varphi \in C(\beta X; Y)$ such that the following diagram commutes:
\[\xymatrix{ \beta X \ar@{->}[r]^{\beta \varphi} & Y \\ X \ar@{->}[u]^{e} \ar@{->}[ru]_{\varphi} & }\]
The pair $(\beta X, e)$ is the Stone-Čech compactification of $X$.
Proof. Let $e: X \to [0, 1]^{C(X; [0, 1])}$ be the embedding of $X$ into $[0, 1]^{C(X; [0, 1])}$ associated with $C(X; [0, 1])$ in Definition 5.24.4, and $\beta X = \ol{e(X)}$.
(1): By Theorem 5.16.6 and Proposition 5.8.3, $[0, 1]^{C(X; [0, 1])}$ is a compact Hausdorff space. By definition, $e(X)$ is dense in $\beta X$.
(U1): For each $f \in C(X; [0, 1])$, $\pi_{f} \in C([0, 1]^{C(X; [0, 1])}; [0, 1])$ is an extension of $f$ to $e(x)$.
(U2): Let $(Y, \varphi)$ be a compactification of $X$. For each $f \in C(Y; [0, 1])$, by (U1), there exists a unique $\beta(f \circ \varphi) \in C(X; [0, 1])$ such that the following diagram commutes:
Let $e': Y \to [0, 1]^{C(Y; [0, 1])}$ be the embedding of $Y$ into $[0, 1]^{C(Y; [0, 1])}$ associated with $C(Y; [0, 1])$, then by (U) of the product topology, there exists $\beta(e' \circ \varphi) \in C(\beta X; [0, 1])$ such that the following diagram commutes:
Since $Y$ is a compact Hausdorff space, $e'(Y)$ is closed by Proposition 5.16.2 and Proposition 5.16.3. As $e'$ is an embedding, identify $Y$ as a subspace of $[0, 1]^{C(Y; [0, 1])}$. Given that $e(X)$ is dense in $\beta X$, the the image of $\beta (e' \circ \varphi)$ lies in $Y$ by Proposition 5.5.3. Therefore under the identification, the following diagram commutes:
$\square$
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