Question about "typical set" in Shannon's source coding theorem
$S_δ$ is not a subset of the typical set. As you mentioned, the most probable element is a member of $S_δ$ but it is not necessarily a member of the typical set.

The only reason to use the typical set instead of $S_δ$ is to make the proof of the source coding theorem easier. (See the paragraph at the top of page 84 in your book.) The typical set along with the asymptotic equipartition property makes it easy to count the size of the typical set and to put bounds on the probability of each element in the set. This is not so easy to do when working with $S_δ$