We show that a locale possesses the localic analogue of the property of realcompactness if and only if it is regular Lindelöf. Thus, the localic version of the Hewitt real-compactification, originally defined by G.Reynolds using σ-frames, is the regular Lindelöf reflection. An immediate consequence is that a space is realcompact if and only if it is the point space of a regular Lindelöf local (3·2). We point out a nice analogy between a theorem of Reynolds and Stone's classical representation theorem for boolean algebras. Finally, we show that the quasi-F cover of a compact Hausdorff space is the Stone–čech compactifications of the smallest dense Lindelöf sublocale.
We present a geometric interpretation of the operation a ⊕ b and the gyration on the unit-disc as defined by A.A. Ungar. Using this geometric interpretation we show that the two known generalizations to the n-dimensional unit ball are identical. The interpretation in the plane leads us to the notion of outer-median of a triangle and we discuss some possible properties of this median. 2004 Elsevier B.V. All rights reserved. MSC: primary 55M30; secondary 54F99, 54H25, 55M10
The concept of an //-set (a generalization of an //-closed space) was introduced by N. V. Velicko. In this paper we obtain internal properties of //-sets in terms of the Itiadis absolute EX and the Hausdorff absolute PX. Some of the main results are:-An //-closed space JΠs Urysohn iff P~ι(A) is an //-set in PXfor every //-set A c X. 1. Introduction. Throughout this paper all spaces are assumed to be Hausdorff. Our main interests in this paper are the //-closed subspaces and the //-sets of a given space X. Our goal is to characterize these subsets. Our first attempt toward such a characterization is to consider preimages of these subsets in the Iliadis absolute EX and the Hausdorff absolute PX. If X is an //-closed Urysohn space we obtain an answer, namely: A c X is an //-set iff P~\A) c PX is an //-set. For //-closed subspaces of an //-closed Urysohn space the situation is less clear. We give an example of such a space X and an //-closed subspace A such that P~\A) is not //-closed.For the case of non-Urysohn //-closed spaces the situation differs. We present an example of an //-set in a space X which is not the image of a compact subset of EX(oτ an //-set in PX).Although our approach to these questions is quite technical, results 4.7, 4.8, 5.1 and 5.7(i) show that it is rather useful. However, many questions remain unanswered.
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.