No abstract

Abstract. In this paper we present an example in the setting of closure spaces that fits in the general theory on 'complete objects' as developed by G. C. L. Brümmer and E. Giuli. For V the class of epimorphic embeddings in the construct Cl0 of T0 closure spaces we prove that the class of V-injective objects is the unique firmly V-reflective subconstruct of Cl0. We present an internal characterization of the Vinjective objects as 'complete' ones and it turns out that this notion of completeness, when applied to the topological setting is much stronger than sobriety. An external characterization of completeness is obtained making use of the well known natural correspondence of closures with complete lattices. We prove that the construct of complete T0 closure spaces is dually equivalent to the category of complete lattices with maps preserving the top and arbitrary joins. 2000

No abstract

Abstract. The purpose of this note is to give necessary and sufficient conditions for a convergence space (X, q) such that for every filter on X its adherence is a closed subset of (X, q). An internal characterization of this property is given by weakening the diagonal condition of Kowalsky. An external characterization is given using a hyperspace structure on the collection of all closed subsets of the given space. It will be shown that a convergence space has closed adhérences if and only if the hyperspace is compact.Introduction. We shall investigate what conditions a convergence space has to fullfill in order that every filter has a closed adherence. In general, adhérences of filters in a convergence space are not closed. This is one of the essential differences between a topological space and a general convergence space. Nevertheless many nontopological convergence spaces are known in which adhérences of filters are closed (for example, almost topological spaces and diagonal convergence spaces).We shall give two characterizations of spaces with closed adhérences. An internal characterization will be obtained by weakening the diagonal condition of Kowalsky. We shall introduce weakly diagonal convergence spaces and show that these are exactly the convergence spaces with closed adhérences. For pretopological spaces the notions almost topological, diagonal and weak diagonal are all equivalent with the idempotency of the closure operator. We show that in general the class of weakly diagonal spaces is strictly larger than the class of diagonal or almost topological spaces but strictly smaller than the class of spaces with an idempotent closure operator.An external characterization for a convergence space to have closed adhérences will be given using a hyperspace structure on the collection of closed subsets of the given space. In fact the structure considered is the Choquet hyperspace described in [2, p. 87] which also corresponds to the convergence of nets of closed sets as defined by Frolik in [4, p. 169] and by Mrówka in [9, p. 59]. The adhérences of filters on a given convergence space are closed if and only if the hyperspace is compact.For all notational conventions and for definitions on convergence spaces

No abstract