= Solution
Suppose first that $T$ is aleph-zero-categorical. If a type in some $S_n(T)$ were nonisolated, the <omitting types theorem> would produce a countable model omitting it, while a countable elementary submodel of a model realizing it would be another countable model. This contradicts categoricity. Thus every type is isolated. The compact Stone space $S_n(T)$ is then discrete and therefore finite.
Conversely, if every $S_n(T)$ is finite, every type is isolated. Every countable model is consequently atomic, and part i says that any two countable models are isomorphic. This proves the <Ryll-Nardzewski theorem>.
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