Some uniformities on X related to topological properties of C(X)C(X)

If X   is a Hausdorff completely regular space, we characterize those spaces Cc(X)Cc(X) whose compact sets are metrizable in terms of a particular uniformity on X. This fact is used to show that for a k-space X are equivalent (i) X   satisfies the discrete countable chain condition, (ii)(ii) every admissible uniformity on X   is trans-separable, and (iii)(iii) every compact set of Cc(X)Cc(X) is metrizable. Several examples examine this result.