Let $m$ denote the Lebesgue-measure on the unit interval and let $L^{\infty}(m)$ be the set of measurable essentially bounded functions on that interval (i.e. $f \in L^{\infty}(m)$ if and only if there exists $\lambda$ such that $|f|\leq \lambda$ almost surely). Now consider the maximal ideal space of $L^{\infty}(m)$.
In general the maximal ideal space is a compact Hausdorff space. I have two questions concerning further topological properties:
1) Is the maximal ideal space in this example separable?
2) Does every point of the above maximal ideal space - i.e. every complex homomorphism - have a countable! fundamental system of neighbourhoods?