
LatticeValued Convergence Spaces: Weaker Regularity and RegularityDOI: 10.1155/2014/328153 Abstract: By using some latticevalued Kowalsky’s dual diagonal conditions, some weaker regularities for J？ger’s generalized stratified convergence spaces and those for Boustique et al’s stratified convergence spaces are defined and studied. Here, the lattice is a complete Heyting algebra. Some characterizations and properties of weaker regularities are presented. For J？ger’s generalized stratified convergence spaces, a notion of closures of stratified filters is introduced and then a new regularity is defined. At last, the relationships between regularities and weaker regularities are established. Dedicated to the first author’s father Zonghua Li on the occasion of his 60th birthday 1. Introduction In 1954, Kowalsky [1] introduced a diagonal condition (the Kdiagonal condition) to characterize whenever a pretopological convergence space is topological. In 1967, Cook and Fischer [2] defined a stronger diagonal condition (the Fdiagonal condition) which, as they showed therein, is necessary and sufficient for a convergence space to be topological. Furthermore, a dual version of F (the DFdiagonal condition) is necessary and sufficient for a convergence space to be regular. Regularity can also be characterized by the requirement that, for each filter , if converges to then so does (the closure of ). In [3, 4], by considering a pair of convergence spaces and , Kent and his coauthors introduced a kind of relative topologicalness (resp., regularity) which was called topologicalness (resp., regularity). They discussed topologicalness (resp., regularity) both by neighborhood (resp., closure) of filter [5] and generalized F (resp., DF)diagonal condition. When , topologicalness (resp., regularity) is precisely topologicalness (resp., regularity). In 1996, Kent and Richardson defined a weaker regularity by using the duality of Kowalsky’s diagonal condition. They also proved that weaker regularity, regularity, and regularity were distinct notions but closely related to each other [6]. In [7], J？ger investigated a kind of latticevalued convergence spaces, which were called generalized stratified convergence spaces. Later, the theory of these spaces was extensively discussed under different lattice context [8–19]. A supercategory of generalized stratified convergence spaces, called levelwise stratified convergence spaces in this paper, was researched in [20–24]. Indeed, a generalized stratified convergence space is precisely a leftcontinuous levelwise stratified convergence space [22]. Latticevalued K and Fdiagonal conditions for generalized stratified
