Stone-Čech compactification ($\beta X$)

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 $\tilde{f}(p)=p(f)$, 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.

Every continuous map from a space to a compact Hausdorff space extends continuously to its Stone-Čech compactification. Uniqueness follows from density of the original space and the Hausdorff property of the target. Two Hausdorff compactifications with this property have comparison maps both ways; their composites fix the dense original space and therefore are identity maps, proving equivalence of the compactifications.

In the Stone-Čech compactification of a discrete space, the closure of each subset of $X$ is clopen, because its indicator extends to a continuous zero-one-valued function. For disjoint closed sets $A,B\subseteq \beta X$, choose $u=0$ on $A$ and $u=1$ on $B$ by Urysohn's lemma. Then $U=\overline{\{x\in X:u(x)<1/2\}}$ and $V=\beta X\setminus U$ are disjoint clopen sets containing $A$ and $B$, respectively. Density proves these containments.

For a discrete space $X$ and $D\subseteq X$, the continuous extension of $1_D$ to $\beta X$ remains idempotent by density, so it takes values in $\{0,1\}$. Its one-set is the clopen closure of $D$. If $O$ is any open set in $\beta X$, density gives $\overline{O}=\overline{O\cap X}$, so this closure is clopen as well. Thus $\beta X$ is an extremally disconnected space. For infinite $X$, this does not imply that every subset of $\beta X$ has open closure.