= Solution
A <uniform algebra> on a <compact Hausdorff space> $K$ is a closed complex subalgebra $A\subseteq C(K)$ containing the constants and separating points of $K$, with the <supremum norm>. It is therefore a commutative unital <Banach algebra>. Its <character space of an algebra> $\Phi_A$, also called its carrier space, consists of nonzero multiplicative <linear functionals>, equipped with the <Gelfand topology>.
For $x\in K$, evaluation gives the <character of an algebra> $\delta_x(f)=f(x)$. The map $\delta:K\to\Phi_A$ is continuous: composing it with the coordinate $\varphi\mapsto\varphi(f)$ gives the continuous function $f$ for every $f\in A$. It is injective because $A$ separates points. The <Gelfand topology> is Hausdorff because distinct <algebra characters> differ on some element of $A$. A continuous injection from compact $K$ into a Hausdorff space is a <homeomorphism> onto its image: closed subsets of $K$ have compact, hence closed, images. Its image is also compact and closed. Thus
$$
\boxed{\delta:K\longrightarrow j(K)=\{\delta_x:x\in K\}\subseteq\Phi_A}
$$
is the asserted <homeomorphism>. A <natural uniform algebra> is one for which this image is all of $\Phi_A$.
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