Insertion of lattice-valued and hedgehog-valued functions

International Journal of Mathematics and Mathematical Sciences, 1989

Let X be a Wilker space and M(X,Y) the set of continuous multifunctions from X to a topological space Y equipped with the compact-open topology. Assuming that M(X,Y) is equipped with the partial order c we prove that (M(X,Y),c) is a topological V-semilattice.

Let Y be a locally compact space, C K (Y) the collection of real-valued continuous functions with compact support, and B L (Y) the set of all Baire func-tions with Lindelöf cozero-set. We show that the embedding C K (Y) B L (Y) of archimedea-groups (or vector lattices) has this universal mapping property: any homomorphism C K (Y) ' ! A, where A has the property (*) is divisible and both conditionally and laterally -complete, has a unique extension B L (Y) ' ! A; also, B L (Y) has property (*). Our perspec-tive on this is as follows. In a category C an object G is epicomplete if the only epic monics out of G are iso-morphisms, epic or monic meant in the categorical sense of right or left cancellable. For each of the categories Arch, archimedea-groups wit-homomorphisms, and its companion category W, Arch-objects with distinguished weak unit and unit-preservin-homomorphisms, (and for the corresponding categories of vector lattices) epicompleteness has been characterized as divi...

Algebra Universalis, 1997

Recent work in the ideal theory of commutative rings and that of C*-algebra is unified and generalized by first noting that these spaces are Lawson-closed subspaces of continuous lattices, equipped with the restriction of the lower topology. These topologies were first studied by Nachbin in the late 1940's (in [32]), as the topologies of those open sets in a compact Hausdorff space which are upper sets with respect to a partial order closed in its square. Such spaces have an intrinsic duality, obtained by reversing the order.¶ We use a different characterization of these spaces, which allows convenient topological proofs: A bitopological space $ (X, \tau, \tau^*)$ is joincompact if $ \tau \vee \tau^* $ is quasicompact and T 0, whenever x is in the $ \tau^* $ -closure of y then y is the $ \tau $ -closure of x, and for any $ x, y \in X $ , if x is not in the $ \tau $ -closure of y, there exist disjoint $ T \in \tau, T^* \in \tau^* $ such that $ x \in T $ and $ y \in T^* $ . Joincompact spaces are shown to be precisely the Lawson-closed subsets of continuous lattices, with the restrictions of the lower and Scott topologies. The natural duality of Nachbin's spaces here takes the form, $ (X, \tau , \tau^*) \rightarrow (X, \tau^*, \tau) $ , but duality is absent in general continuous lattices. Duality permits efficient proofs of very general results on compactness, the Baire property, and coincidence of topologies, for maximal and minimal elements in Lawson-closed subspaces of continuous lattices.

Fundamenta Mathematicae

We study the class of perfect images of generalized ordered (GO) spaces, which we denote by PIGO. Mary Ellen Rudin's celebrated result characterizing compact monotonically normal spaces as the continuous images of compact linearly ordered spaces implies that every space with a monotonically normal compactification is in PIGO. But PIGO is wider: every metrizable space is in the class, but not every metrizable space has a monotonically normal compactification. On the other hand, a locally compact space is in PIGO if and only if it has a monotonically normal compactification. We answer a question of Bennett and Lutzer that asked whether a (semi)stratifiable space with a monotonically normal compactification must be metrizable by showing that any semistratifiable member of PIGO is metrizable. This also shows that there are monotonically normal spaces which are not in PIGO. We investigate cardinal functions in PIGO, and in particular show that if K is a compact subset of a space X in PIGO, then the character of K in X equals the pseudo-character of K in X. We show that the product of two non-discrete spaces in PIGO is not in PIGO unless both are metrizable or neither one contains a countable set with a limit point. Finally, we look at the narrower class of perfect images of linearly ordered spaces, which we denote by PILOTS. Every metrizable space is in PILOTS, and if a space in PILOTS has a G δ-diagonal, then it must be metrizable. Thus familiar GO-spaces such as the Sorgenfrey line and Michael line are in PIGO but not in PILOTS.

Fuzzy Sets and Systems, 2015

This paper provides variable-basis lattice-valued analogues of the well-known results that the construct Prost of preordered sets, firstly, is concretely isomorphic to a full concretely coreflective subcategory of the category Top of topological spaces (which employs the concept of the dual of the specialization preorder), and, secondly, is (non-concretely) isomorphic to a full coreflective subcategory of the category TopSys of topological systems of S. Vickers (which employs the spatialization procedure for topological systems). Dualizing these results, one arrives at the similar properties of quasi-pseudo-metric spaces built over locales.