An internal and an external characterization of convergence spaces in which adherences of filters are closed

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

Open and Proper Maps Characterized by Continuous Setvalued Maps

“…Suppose/ is continuous and let J^~ be a filter on X. There exists an ultrafilter x on &(X) such that ^(x) = ^~-The proof of this statement is similar to (2.7) and can be found in [8]. x converges to a(J^) and so/(x) converges to/(a(J r )).…”

“…Let J^(x) be the filter generated. Then sup x is the adherence of ^~(x)> which is denoted by aJ^(x) [8], [9]. The infimum, inf x is the set of points p (z X with the property that for each neighborhood V of p there exists an sé Ç x such that for each A ^ se, A C\ V ^ ÏÏ [2], [5].…”

Non-Hausdorff convergence spaces

“…Suppose (1) and (2) hold. Then X is pretopological by Proposition 11. so, by Theorem 1.4 of [5], X is topological if and only if it is diagonal. But condition (1) is precisely the diagonal condition in X translated back to X using Proposition 8.…”

“…The reader is referred to [5] for information on diagonal theorems and their relation to topologies and to [2] and [4] for a general discussion of quotient spaces.…”

Three problems in convergence theory

“…In [15], Eva Lowen-Colebunders introduced and investigated convergences ξ such that every filter has a closed adherence, that is, such that adh ξ (adh ξ F ) = adh ξ F for every filter F on |ξ|. She formulated a condition, called weak diagonality, which is a weakening of the diagonality property of Kowalsky and proved that a convergence is weakly diagonal if and only if filters have closed adherences.…”