Some uniformities on X related to topological properties of

If X is a Hausdorff completely regular space, we characterize those spaces C c ( 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, ( i i ) every admissible uniformity on X is trans-separable, and ( i i i ) every compact set of C c ( X ) is metrizable. Several examples examine this result.