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Lattice-Valued Convergence Spaces: Weaker Regularity and -Regularity

DOI: 10.1155/2014/328153

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By using some lattice-valued 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 K-diagonal condition) to characterize whenever a pretopological convergence space is topological. In 1967, Cook and Fischer [2] defined a stronger diagonal condition (the F-diagonal condition) which, as they showed therein, is necessary and sufficient for a convergence space to be topological. Furthermore, a dual version of F (the DF-diagonal 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 lattice-valued 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 left-continuous levelwise stratified -convergence space [22]. Lattice-valued K- and F-diagonal conditions for generalized stratified


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