For certain classes of Priifer domains A, we study the completion A'~-of A with respect to the supremum topology3= sup{~lw ~ ~/, where l~ is the family of nontrivial valuations on the quotient field which are nonnegative on A and ~ is a topology induced by a valuation w ~ IL It is shown that the concepts "SFT Priifer domain" and "generalized Dedekind domain" are the same. We show that if E is the ring of entire functions, then E'J is a Bezout ring which is not a ff:-PriJfer ring, and if A is an SFT Priifer domain, then .~.3-is a Priifer ring under a certain condition. We also show that under the same conditions as above, ~,9" is a ff:-PriJfer ring if and only if the number of independent valuation overrings of A is finite. In particular, if A is a Dedekind domain (resp., h-local Priifer domain), then 3-A' is a J--Priifer ring if and only if A has only finitely many prime ideals (resp., maximal ideals). These provide an answer to Mockor's question.