5.25 Compactifications
Definition 5.25.1 (Compactification).label Let $X$ be a topological space, then a compactification of $X$ is a pair $(Y, f)$ where
- (1)
$Y$ is a compact Hausdorff space.
- (2)
$f \in C(X; Y)$ is an embedding.
- (3)
$f(X)$ is dense in $Y$.
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 (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 5.16.8 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 5.16.3 and Proposition 5.16.4. 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:
$\square$
Lemma 5.25.3.label Let $X$ be an LCH space and $(Y, \varphi)$ be a compactification of $X$, then $\varphi(X) \subset Y$ is open.
Proof. For each $x \in X$, let $U \in \cn_{X}(x)$ be a compact neighbourhood. Since $Y$ is a compact Hausdorff space, $\varphi(U)$ is closed by Proposition 5.16.4. As $\varphi \in C(X; Y)$ is an embedding, there exists $V \in \cn_{Y}(\varphi(x))$ such that $\varphi(U) = \varphi(X) \cap V$. Given that $\varphi(X)$ is dense in $Y$, $\varphi(U) = \ol{\varphi(X) \cap V}\supset V$. Therefore $\varphi(U) \in \cn_{Y}(\varphi(x))$, and $\varphi(X)$ is open in $Y$.$\square$
Definition 5.25.4 (One-Point Compactification).label Let $(X, \mathcal{T})$ be an LCH space, then there exists a pair $(X^{*}, \iota)$ such that:
- (1)
$(X^{*}, \iota)$ is a compactification of $X$.
- (U)
For any pair $(Y, \varphi)$ satisfying (1), there exists a unique $\varphi^{*} \in C(Y; X^{*})$ such that the following diagram commutes:
\[\xymatrix{ Y \ar@{->}[rd]^{\varphi^*} & \\ X \ar@{->}[r]_{\iota} \ar@{->}[u]^{\varphi} & X^* }\]
The pair $(X^{*}, \iota)$ is the one-point compactification of $X$.
Proof. Let $\infty$ be a point not in $X$, $X^{*} = X \sqcup \bracs{\infty}$, and $\mathcal{T}^{*} \subset 2^{X^*}$ such that for each $U \in \mathcal{T}^{*}$, one of the following holds:
- (a)
$U \in \mathcal{T}$.
- (b)
$\infty \in U$ and $U^{c} \subset X$ is compact.
Let $\seqi{U}\subset \mathcal{T}^{*}$ be an open cover of $X$, then there exists $i \in I$ such that $\infty \in U$. In which case, $U_{i}$ must satisfy (b), so there exists $J \subset I$ finite such that $\bigcup_{j \in J}U_{j} \supset U_{i}^{c}$, and $\bracsn{U_j|j \in J \cup \bracs{i}}$ is a finite subcover. Now, let $x \in X$, then since $X$ is locally compact, there exists a relatively compact neighbourhood $U \in \cn_{X}^{o}(x)$. In which case, $\ol{U}^{c} \in \cn_{X^*}(\infty)$ with $U \cap \ol{U}^{c} = \emptyset$. Therefore $X^{*}$ is a compact Hausdorff space.
Let $\iota: X \to X^{*}$ be the inclusion map. For each $U \in \mathcal{T}^{*}$ satisfying (b), $\iota^{-1}(U) = U \cap X$. Since $U^{c} \subset X$ is compact, $U \cap X$ is open by Proposition 5.16.4, so $\iota \in C(X; X^{*})$. By (a), $\iota$ is an embedding.
Finally, let
Let $U \subset X^{*}$ with $\infty \not\in U$, then $(\varphi^{*})^{-1}(U) = \varphi(U)$ is open in $\varphi(X)$ because $\varphi$ is an embedding, and open in $Y$ by Lemma 5.25.3. On the other hand, for each $V \in \cn_{X^*}^{o}(\infty)$,
Since $\varphi \in C(X; Y)$ is an embedding, $V$ is relatively open in $\varphi(X)$, so $V \cup (Y \setminus \varphi(X))$ is open in $Y$.$\square$
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