Theorem 16.9 (Ryll-Nardsewski / Engeler / Svenonius 59). Assuming that:

Then the following are equivalent:

(i) $T$ is $ℵ_{0}$ -categorical.
(ii) For all $n \geq 1$ , every type in $S_{n} (T)$ is isolated.
(iii) For all $n \geq 1$ , $S_{n} (T)$ is finite.
(iv) For all $n \geq 1$ , the number of $L$ -formulas with $x_{1}, \dots, x_{n}$ free variables is finite, modulo $T$ .