= Stone-Čech compactification
{c}
{title2=$\beta X$}
For a <completely regular Hausdorff space>, the Stone-Čech compactification is the weak-star closure of the evaluation functionals in the dual unit ball of $C_b(X)$. The <Banach-Alaoglu theorem> gives compactness; complete regularity makes evaluation a <topological embedding>. Every bounded real <continuous function> extends by $\widetilde f(p)=p(f)$, yielding an isometric isomorphism $C_b(X)\cong C(\beta X)$. For a continuous map into a <compact Hausdorff space> $K$, the adjoint of pullback sends this closure into the closed evaluation copy of $K$, yielding the unique continuous extension. Uniqueness follows from density. It is a universal compactification, not a homeomorphism between $X$ and the whole dual space.
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