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).
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.