We consider $A = C_{b}(X)$, the ring of continuous bounded functions on a completely regular space $X$. Let $\DeclareMathOperator{\SpecMax}{SpecMax} \SpecMax(A)$ be the set of maximal ideals of $A$ with the Zariski topology.

We know that there is an embedding of topological spaces: $$ \psi : X \longrightarrow \SpecMax(A) $$ defined by $\psi(x) = m_{x} := \lbrace f\in A \mid f(x)=0\rbrace$.

My question is: We can construct the Stone–Čech compactification of $X$ if we take $\bar{X}:=\overline{\psi(X)}$, but we must prove that $\overline{\psi(X)}$ is compact. We know that $\operatorname{Spec}(A)$ is quasi-compact, but I don't know if $\SpecMax(A)$ is also. Then how can we prove that $\overline{\psi(X)}$ is compact?

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