A regular convergence space has
both a finest and coarsest compatible regular Cauchy structure. The coarsest
compatible regular Cauchy structure is complete if and only if the original space is
Urysohn-closed; it is totally bounded if and only if the original space is almost
topological. Minimal regular Cauchy spaces are characterized and, in the complete
case, shown to be in one-to-one correspondence with the minimal regular convergence
spaces. The noncomplete minimal regular regular Cauchy spaces do not have regular
completions.
