$E^*$-Stable Theories

S. Shelah proved that stability of a theory is equivalent to definability of every complete type of that theory. T. Mustafin introduced the concept of being $T^*$-stable, generalizing the notion of being stable. However, $T^*$-stability does not necessitate definability of types. The key result of the present article is proving the definability of types for $E^*$-stable theories. This concept differs from that of being $T^*$-stable by adding the condition of being continuous. As a consequence we arrive at the definability of types over any $P$-sets in $P$-stable theories, which previously was established by T. Nurmagambetov and B. Poizat for types over $P$-models.