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Let $\mathcal{F}=C(X,[0,1])$. Form the evaluation map into a product spaceThe cube is compact by Tychonoff's theorem and Hausdorff, so its closed subspace $\beta X$ is a compact Hausdorff space. A completely regular Hausdorff space has enough continuous functions to separate points and closed sets. Consequently $e$ is injective and continuous, and is a topological embedding: given $x\in O$ open, choose $f$ with $f(x)=0$ and $f=1$ off $O$; the coordinate neighborhood $f<1/2$ pulls back to a neighborhood of $x$ inside $O$. Identify $X$ with this dense embedded subspace. This constructs the Stone-Čech compactification.

Each $f\in \mathcal{F}$ extends as its coordinate function. Rescaling therefore extends every bounded real continuous function; real and imaginary parts extend bounded complex continuous functions. The extension is unique because $X$ is dense and the target is Hausdorff. Its supremum norm is unchanged by density, so restriction is an isometric isomorphism of normed spaces of Banach spacesThese are the space of continuous functions on a compact space and the space of bounded continuous functions, respectively.

More generally, embed a compact Hausdorff space $K$ in its own evaluation cube $[0,1{]}^{C(K,[0,1])}$. This is an embedding by complete regularity, and its image is closed by compactness. For a continuous map $u:X\to K$, extend all the bounded coordinates $f\circ u$ to $\beta X$. The resulting continuous map into the cube takes its values in the closed copy of $K$: it does so on the dense subset $X$, and the inverse image of that closed copy is closed. ThusUniqueness again follows from density. This is the universal property of the Stone-Čech compactification. It implies uniqueness of the compactification up to a homeomorphism fixing $X$, by extending the identity in both directions. Every Hausdorff compactification of $X$ is the image of $\beta X$ under a continuous surjection fixing $X$, since that image is compact and contains the dense copy of $X$. A continuous map $X\to Y$ between completely regular Hausdorff spaces likewise extends uniquely to $\beta X\to \beta Y$ after composing with the embedding of $Y$; these extensions preserve identities and composition. If $X$ is already compact, its dense copy is closed, so $\beta X=X$.

For discrete $X$, every indicator $1_D$, $D\subseteq X$, is continuous. Its extension to $\beta X$ takes only values $0,1$, since this closed condition holds on the dense subset $X$. Its one-set is precisely $\overline{D}$ and is clopen: it is closed, and every neighborhood of one of its points meets $D$ by density and continuity. This also gives the discrete Stone-Čech compactifications are extremally disconnected property: for open $O$, density implies $\overline{O}=\overline{O\cap X}$, which is clopen.

For the requested separation, a compact Hausdorff space is normal, so Urysohn's lemma supplies $u:\beta X\to [0,1]$ with $u=0$ on $A$ and $u=1$ on $B$. Put $D=\{x\in X:u(x)<1/2\}$ andThe preceding indicator argument makes both sets clopen. At a point of $A$, the neighborhood $u<1/2$ has its dense $X$-part inside $D$, so the point lies in $U$. At a point of $B$, the neighborhood $u>1/2$ misses $D$, so the point lies outside $U$. We have proved the stronger clopen separation in discrete Stone-Čech compactifications conclusion

Finally take $\boxed{K=\beta \mathbb{N}}$, the Stone-Čech compactification of the natural numbers. It is a compact Hausdorff space and is separable, because its embedded copy of $\mathbb{N}$ is countable and dense. For every $D\subseteq \mathbb{N}$, the indicator extends to an element $h_D\in C(K)$. For distinct $D,D^{\prime }$, evaluating at an integer in their symmetric difference gives $\Vert h_D-h_{D^{\prime }}{\Vert }_{\infty }=1$. There are uncountably many such indicators. A separable metric space cannot contain an uncountable family at pairwise distance one: disjoint balls of radius $1/3$ would require distinct members of a countable dense set. Hence $C(K)$ is not separable, despite separability of $K$.

A completely regular Hausdorff space is a Hausdorff space in which each point $x$ outside a closed set $F$ is separated from $F$ by a continuous function $u:X\to [0,1]$ with $u(x)=1$ and $u{|}_F=0$. Put $E=C_b(X)$ and define the evaluation character by ${\delta }_x(g)=g(x)$. The weak-star topology on $E^{\prime }$ is the topology of pointwise convergence on $E$: its coordinate maps $p\mapsto p(g)$ are continuous, and generate the topology. It is Hausdorff, and the dual unit ball is compact in it by the Banach-Alaoglu theorem.

For nonempty $X$, $\Vert {\delta }_x\Vert =1$ by the supremum norm bound and the constant function $1$. All the coordinates of $x\mapsto {\delta }_x$ are continuous, so $\delta$ is continuous. Complete regularity separates distinct points, proving injectivity. To prove continuity of the inverse onto its image, let $x\in O$ with $O$ open in $X$. Separating $x$ from $X\setminus O$ gives a bounded continuous $u$ withSuch neighborhoods prove that $\delta$ is a homeomorphism onto $\delta (X)$. This qualification is essential: it is not onto the whole dual, since the zero functional is not an evaluation on a nonempty space.

Define the Stone-Čech compactification to beThe Banach-Alaoglu theorem makes this a compact Hausdorff space, with $X$ identified with its dense evaluation copy. For $g\in C_b(X)$, the coordinate function $\tilde{g}(p)=p(g)$ is a continuous extension. Density gives $\Vert \tilde{g}{\Vert }_{\infty }=\Vert g{\Vert }_{\infty }$. Conversely any $h\in C(\beta X)$ restricts to an element $g$ of $C_b(X)$, and $h=\tilde{g}$ because they agree on the dense subset $X$. Restriction and coordinate extension are inverse linear isometries, henceThey also preserve products: the corresponding continuous functions agree on $X$ and hence everywhere.

For the universal extension property, let $f:X\to K$ be continuous and $K$ a compact Hausdorff space. Pullback defines a bounded linear operator $T:C(K)\to C_b(X)$ by $Tg=g\circ f$. Its dual map $T^{\prime }:E^{\prime }\to C(K{)}^{\prime }$ is weak-star continuous, since $(T^{\prime }p)(g)=p(Tg)$. Evaluation ${\delta }_K:K\to C(K{)}^{\prime }$ is a homeomorphism onto a compact, thus closed, subset: a compact Hausdorff space is completely regular, by the Urysohn lemma. Now $T^{\prime }{\delta }_x={\delta }_K(f(x))$, soTherefore $\overline{f}={\delta }_K^{-1}\circ T^{\prime }{|}_{\beta X}$ is the required continuous extension. Two continuous maps into the Hausdorff space $K$ agreeing on the dense subset $X$ agree everywhere, proving uniqueness.

If $X$ is open in $\beta X$, then it is a locally compact space: around any $x$, regularity of the compact Hausdorff space $\beta X$ gives an open neighborhood whose closure is compact and lies in $X$. Conversely, suppose $X$ is locally compact. Choose an open neighborhood $U$ of $x$ in $X$ with compact closure $C\subseteq X$. There is an open $O\subseteq \beta X$ with $O\cap X=U$. Density of $X$ implies $O\subseteq {\overline{U}}^{\beta X}$, while compactness of $C$ makes it closed in $\beta X$, so ${\overline{U}}^{\beta X}\subseteq C$. Thus $x\in O\subseteq X$. This provesThe empty-space cases are immediate.

The final printed assertion, for an arbitrary subset of $\beta X$, is false. Take the infinite discrete space $X=\mathbb{N}$. It is not compact, so its compactification has a point $p\in \beta \mathbb{N}\setminus \mathbb{N}$. The singleton $\{p\}$ is closed, hence equals its own closure. It is not open, since every nonempty open subset of $\beta \mathbb{N}$ meets the dense copy of $\mathbb{N}$. This is a counterexample.

The valid result holds for subsets $D\subseteq X$, and more generally for open subsets of $\beta X$. For $D\subseteq X$, its indicator is bounded continuous because $X$ is discrete. Its extension $\widetilde{1_D}$ is still idempotent by density, so it takes only the values zero and one. The set $H=\{\widetilde{1_D}=1\}$ is clopen and has $H\cap X=D$. Density then gives $H={\overline{D}}^{\beta X}$. If $O$ is open in $\beta X$, the set $O\cap X$ is dense in $O$, so $\overline{O}=\overline{O\cap X}$ is clopen by the result just proved. Consequently $\beta X$ is extremally disconnected, which means the closure of every open set is open. This is the discrete Stone-Čech compactifications are extremally disconnected theorem, with the essential open-set qualification.

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.