Uniformities,hyperspaces, and normality

LetX be a completely regular space and 2^X the hyperspace ofX. It is shown that the uniform topologies on 2^X arising from Nachbin uniformity onX, which is the weak uniformity generated byC(X, ), and from Tukey—Shirota uniformity onX, generated by all countable open normal coverings ofX, agree. They, both, coincide with a Vietoris-type topology on 2^X, the countable locally finite topology, iffX is normal.