Let (X,d) be a metric space with metric d. This paper studies Wijsman convergence (w-convergence for short) in 2 x, the power set of X, and \(2_ 0\) X the subfamily of non-empty subsets of X, as well as the hyperspace c(X), which is the set of all closed subsets of X and \(C_ 0(X)\) the set of all non-empty closed subsets of X. The paper first describes uniformities U and V for 2 2 and \(2\) \(x_ 0\) which are compatible with w-convergence on X and proceed to describe: an explicit pseudo-metric \(\alpha_ x\) on 2 x for each \(x\in X\) and shows that (1) U is generated by the continuous family of pseudo-metric \(\{\alpha_ x:\) \(x\in X\}\), (2) (2 x,U) is totally bounded, (3) \((2\) \(X_ 0,V)\) is totally bounded iff d is totally bounded and lastly, (4) if X is a separable metric space, then \[ d(A,B)=\sum^{\infty}_{k=1}\frac{1}{2\quad k}\alpha_ k(A,B) \] is a metric on C(X), where \(\alpha_ k\) is the continuous pseudo-metric as in (1) above corresponding to the element \(x_ k\) of a countable dense set \(\{x_ k\}\) in X, and \(\alpha\) generates the uniformity U. Two explicit metrics \(\beta\) and \({\tilde \beta}\), which are compatible with the uniformity V on \(C_ 0(X)\), for a separable metric space X, are also defined and the following three are shown to be equivalent (1) \((C_ 0(X),{\tilde \beta})\) is boundedly compact, (2) \((C_ 0(X),V)\) is complete, (3) (X,d) boundedly compact.