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. 



Assumptions on a commutative C*-algebra to get a nice C(X) - space
The space $X$ is second countable if and only if $C(X)$ is separable for the norm. This is proved, for example, as Theorem 2.4 of the little article ["Notes on the Separability of C\* algebras" by Chun-Yen Chou](http://journal.taiwanmathsoc.org.tw/index.php/TJM/article/view/1786). Actually, the short proof given there works also for locally compact Hausdorff spaces and therefore non-unital $C^{\*}$-algebras.


