A simple Efimov space with sequentially-nice space of probability measures

Under Jensen’s diamond principle , we construct a simple Efimov space K whose space of nonatomic probability measures P_na(K) is first-countable and sequentially compact. These two properties of P_na(K) imply that the space of probability measures P ( K ) on K is selectively sequentially pseudocompact. We show that any sequence of probability measures on K that converges to a purely atomic measure converges in norm, and any sequence of probability measures on K converging to zero in sup-norm has a subsequence converging to a nonatomic probability measure. We show also that the Banach space C ( K ) of continuous functions on K has the Gelfand–Phillips property but it does not have the Grothendieck property.