= Solution
We work with the usual infinite-model convention for an <omega-categorical theory>: it has a unique countably infinite model up to isomorphism. The three conditions are the type-space and automorphism formulations of the <Ryll-Nardzewski theorem>; the following arguments prove them rather than invoke that theorem. The standard hypothesis that $T$ has infinite models is understood. If finite models are allowed instead, uniqueness among models of size at most countable is the matching convention for condition (i).
We first prove (i) implies (ii). Suppose some <type space> $S_n(T)$ were infinite. An infinite compact space cannot have every point isolated: its cover by singleton open sets would have no finite subcover. Thus it has a nonisolated <complete type> $p$. The <omitting types theorem> gives a countable model omitting $p$. On the other hand, <compactness theorem> gives a model realizing $p$, and the <Downward Lowenheim-Skolem theorem> gives a countable elementary submodel containing its realizing tuple. These countable models cannot be isomorphic, since an isomorphism preserves realization of $p$. This contradicts omega-categoricity. Hence \b[every $S_n(T)$ is finite].
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