> [!theorem] > > Let $X$ be a [[Completely Regular Space|completely regular space]], $\cf \subset C(X, [0, 1])$ be a family of [[Space of Continuous Functions|continuous functions]] that separate points and [[Closed Set|closed sets]], $e: X \to I^\cf$ be the associated [[Embedding in Cube|embedding]], and $Y = \ol{e(X)}$ be the [[Topological Closure|closure]] of the image. Then > 1. $(Y, e)$ is a [[Compactification|compactification]] of $X$. > 2. Let $\alg \subset BC(X)$ be the [[Closed Set|closed]] [[Algebra over Ring|subalgebra]] generated by $\cf$, then every $f \in \alg$ has a [[Continuity|continuous]] extension to $Y$. > > *Proof*. By identifying $X = e(X)$, each $f \in \cf$ can be identified with $\pi_\cf$, which yields the extension to $Y$. Uniqueness comes from the density of $X$. > > Let $\cf \subset \cb \subset BC(X)$ be the space of functions that admit a continuous extension to $Y$. By continuity, $\cb$ is an algebra. If $\seq{f_n} \subset \cb$ converges [[Uniform Convergence|uniformly]] to $g \in BC(X)$, then the uniform limit of their extensions converges to a desired extension. > [!definitionb] Stone-Čech Compactification > > > $ > \begin{CD} > X @>{e}>> \beta X @>{\iota}>> I^\cf \\ > @V{\phi}VV @V{\Phi|_{\beta X}}VV @V{\Phi}VV \\ > Y @>>{i}> \beta Y @>>{\iota}> I^{\mathcal G} > \end{CD} > $ > > Let $X$ be a completely regular space, $\cf = C(X, [0, 1])$, $e: X \to I^\cf$ be the associated embedding, and $\beta X = \ol{e(X)}$. Let $Y$ be a [[Compactness|compact]] [[Hausdorff Space|Hausdorff space]] and $\phi \in C(X, Y)$, then > 1. There exists a unique continuous extension $\td \phi \in C(\beta X, Y)$ such that $\td \phi \circ e = \phi$. > 2. If $(Y, \phi)$ is a compactification of $X$, then $\tilde \phi$ is surjective. > 3. If $(Y, \phi)$ is a compactification such that every $f \in BC(X)$ extends continuously to $Y$, then $\tilde \phi$ is a [[Homeomorphism|homeomorphism]]. > > and the pairing $(\beta X, e)$ is known as the **Stone-Čech compactification** of $X$. > > *Proof*. Let $\mathcal G = C(Y, I)$ and let $(\beta Y, i)$ be the Stone-Čech compactification of $Y$. Let $\phi \in C(X, Y)$, define > $ > \Phi: I^\cf \to I^{\mathcal G} \quad \pi_g \circ \Phi = \pi_{g \circ \phi} > $ > then $\Phi$ is continuous since its projection into each coordinate is continuous. The above definition is possible since $g \circ \phi \in BC(X)$. Moreover, > $ > \pi_g \circ \Phi \circ e = \pi_{g\circ \phi} \circ e = g \circ \phi = \pi_g \circ i \circ \phi > $ > so $\Phi \circ e = i \circ \phi$. Therefore $\Phi \circ e(X) = i \circ \phi(X) \subset \beta Y$, and $\Phi(\beta X) \subset \ol{\beta Y} = \beta Y$. > > Let $\td \phi = i^{-1} \circ \Phi|_{\beta X}$, then $\td \phi \circ e = i^{-1} \circ \Phi \circ e = \phi$. Uniqueness comes from the density of $e(X)$ in $\beta X$. If $(Y, \phi)$ is a compactification, then $\phi(X)$ is dense in $Y$, with $\td \phi(\beta X)$ dense in $Y$ and compact (closed), so $\td\phi(\beta X) = Y$. > > Lastly, suppose that every $f \in BC(X)$ is of the form $g \circ \phi$ with $g \in BC(Y)$, then for any $p, q \in I^\cf$, $\Phi(p) = \Phi(q)$ implies that > $ > \pi_f(p) = \pi_{g \circ \phi}(p) = \pi_g \circ \Phi(p) = \pi_g \circ \Phi(q) = \pi_{g \circ \phi}(q) = \pi_f(q) > $ > for all $f \in BC(X)$. Therefore $\Phi$ is bijective, and its restriction to $\beta X$ is a homeomorphism.