Gelfand topology of maximal ideal space is restriction of weak-star topology of dual space

Question

This statement is made in Rudin's Functional Analysis (Theorem 11.9(c) proof). I do not follow because the maximal ideal space is a smaller set than the dual space, so in theory, the Gelfand topology can be coarser (i.e., there are evaluation maps to make continuous). Am I missing something? (Spoiler: I must be.)

One clarification: regardless of the statement, the proof works because compactness in finer means compactness in coarser, but it's that particular statement I do not follow.

Answer 1

Actually, this is much easier to see if you consider how the basic open sets would be defined in both spaces, so answering my own question.