I have the following question,

Is it possible to get somehow a compact Hausdorff space $X$ which is second-countable from a unital commutative C*-algebra. If it is possible, what should we assume for our C*-algebra. Gelfand-Naimark theorem gives us $C(X)$, where $X$ is a compact Hausdorff space, but I'm asking how to get it with the second axiom of countability. Thank you in advance for any answers or hints.