Uniformities, hyperspaces, and normality (Q923408)

Let X be a completely regular space and \(2^ X\) the hyperspace of X. It is shown that the uniform topologies induced on \(2^ X\), via the Hausdorff uniformity, from the Nachbin uniformity, which is the weak uniformity generated by C(X,R), and from the Tukey-Shirota uniformity, generated by all countable open normal coverings of X, agree. They both coincide with a Vietoris-type topology on \(2^ X\), the countable locally finite topology, iff X is normal.