A regular completion with the universal property is obtained for each member of a certain class of Cauchy spaces by embedding the Cauchy space in a complete function algebra with the continuous convergence structure.convergence in C{X) ", Pacific J. Math, (to appear).

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The purpose of this note is to introduce a convergence structure hit) on the collection C(£) of nonempty, compact subsets of a Hausdorff convergence space (£, t). It is shown that if (£, /) is topological, then h(t) agrees with the Vietoris topology on C(£). It is proved that (C(£), h(t)) is Hausdorff, that it inherits regularity from (£, /) and that it is compact whenever (£, t) is compact and regular.

In [7] Richardson constructed a Stone-Čech type compactification R(E) of a Hausdorff convergence space E. Two questions arise in this regard. First, when is R(E) homeomorphic to β(E), β(E) the topological Stone-Čech compactification of E, for a Tychonoff topological space E? Second, if E is a regular convergence space, when is R(E) regular? The last question is motivated by the study of regular compactifications in [6]. In section 2 it will be shown that a necessary and sufficient condition in answer to both questions, is that α = cl(α) for each nonconvergent ultrafilter α on E.

In this paper we will establish a method of removing the Hausdorff assumption from certain convergence space theorems. As specific applications the precise form of the closure of a compact set in a regular non-Hausdorff space is given and the exact relationship between cl and cl 2 in a non-Hausdorff compact regular space is obtained. Necessary and sufficient conditions that the transition space for this procedure be topological or pretopological are found and a few embedding theorems are obtained.

ABSTRACT.A compactification for any given convergence space is constructed. The compactification has the unique continuous extension property subject to reasonable assumptions on the range space. Properties of the compactification are studied.

In this paper the study of minimal P uniform convergence spaces is initiated by investigating minimal Hausdorff uniform convergence spaces and minimal uniformly regular uniform convergence spaces.

ABSTRACT. It is proved that a Hausdorff, totally bounded Completion Axiom space is a uniform space.The method of proof shows that Hausdorff