Gunther Jäger

A CATEGORY OF L-FUZZY CONVERGENCE SPACES

Abstract In this paper we take convergence of stratified L-filters as primitive notion and construct in this way a cartesian closed category, which contains the category of stratified L-topological spaces as reflective subcategory. The class of spaces with non-idempotent stratified fuzzy interior operator is characterized as subclass of the class of our stratified L-fuzzy convergence spaces and a first characterization, which fuzzy convergences stem from stratified L-topologies is established.

Pretopological and topological lattice-valued convergence spaces

We show that the classical axiom which characterizes pretopological convergence spaces splits into two axioms in the general Heyting algebra-valued case. Furthermore, we present a generalization of Kowalskis diagonal condition to the lattice-valued case.

Fischer's Diagonal Condition for Lattice-Valued Convergence Spaces

We study a generalization of a diagonal condition which classically ensures that a convergence space is topological. We show that only under an additional condition, which classically is always true, the validity of this diagonal condition implies that a Heyting algebra-valued convergence space is L-topological.

STRATIFIED L -UNIFORM CONVERGENCE SPACES

We define a supercategory of the category of stratified L-uniform spaces, where L is a complete Heyting algebra. Our category is topological over SET and cartesian closed. In the case where L is the two-point chain, it coincides with the category of uniform convergence spaces. Moreover, we describe the induced stratified L-convergence and show that in case of a stratified L-uniform space this L-convergence is a stratified L-topology.

Lattice-valued convergence spaces: Extending the lattice context

We define a category of stratified L-generalized convergence spaces for the case where the lattice is an enriched cl-premonoid. We then investigate some of its categorical properties and those of its subcategories, in particular the stratified L-principal convergence spaces and the stratified L-topological convergence spaces. For some results we need to introduce a new condition on the lattice (which is always true in the case where the lattice is a frame, but not always true in the more general case). As examples where we may apply the more general lattice context we examine the stratified L-topological spaces and probabilistic limit spaces. We show that the category of stratified L-topological spaces is a reflective subcategory of our category and that the category of probabilistic limit spaces under a T-norm is both a reflective and a coreflective subcategory of our category if we choose the lattice context appropriately.

Pointwise convergence, continuous convergence and even continuity in FNS

We discuss the notions of pointwise convergence, continuous convergence and even continuity, which were previously defined for fuzzy convergence spaces using prefilters, in fuzzy neighborhood spaces and show that these concepts can there be described entirely using ordinary filters.

A common framework for lattice-valued uniform spaces and probabilistic uniform limit spaces

We study a category of lattice-valued uniform convergence spaces where the lattice is enriched by two algebraic operations. This general setting allows us to view the category of lattice-valued uniform spaces as a reflective subcategory of our category, and the category of probabilistic uniform limit spaces as a coreflective subcategory.

Lattice-Valued Cauchy Spaces and Completion

Abstract We define lattice-valued Cauchy spaces. The category of these spaces is topological over SET and cartesian closed. Special examples are lattice-valued uniform convergence spaces and probabilistic Cauchy spaces. We further define completeness and give a completion for these spaces which has the property that Cauchy continuous mappings between spaces can be extended to Cauchy-continuous mappings between their completions.

Fuzzy uniform convergence and equicontinuity

Abstract In our earlier papers we investigated function space structures in FTS, the category of fuzzy topological spaces and in FCS, the category of fuzzy convergence spaces. This paper is devoted to the study of function space structures in FUS, the category of fuzzy uniform spaces due to Lowen. We develop the basic theory and give examples of such structures such as pointwise and compact convergence. We consider separation axioms in these fuzzy function spaces and show that the fuzzy uniformity of uniform convergence is conjoining for C ( X , Y ). We define a notion of equicontinuous sets of mappings on which pointwise and compact convergence coincide.

Compactification of lattice-valued convergence spaces

We define compactness for stratified lattice-valued convergence spaces and show that a Tychonoff theorem is true. Further a generalization of the classical Richardson compactification is given. This compactification has a universal property.