Marta Macho Stadler

t -prefilter theory

Abstract In the first part of this paper, the notion of t -prefilter is introduced and studied, as are the applications λ t and ω t , their properties and relations between them. We characterize the minimal prime t -prefilters finer than a given t -prefilter, by using ultrafilters on X : and we give equivalents conditions of the t -maximality. In the second part, we define t -convergence, and we obtain a characterization of t -compactness (Gantner-Steinlage) by means of the maximal t -prefilters. Finally, some applications of these concepts, related to the continuity, separation axioms and fuzzy compactness are studied.

Strong separation and strong countability in fuzzy topological spaces

Given a fuzzy topological space (X, δ), we introduce new notions of fuzzy separation and fuzzy countability, using the family of its level-topologies: {ιt(σ): t ϵ [0, 1)}. We check that these are well-defined fuzzy topological concepts and we compare them with the analogous fuzzy ones introduced in the literature. We verify that these notions are not equivalent, and we give a large number of examples which illustrate this fact.

Transversely Cantor laminations as inverse limits

We demonstrate that any minimal transversely Cantor compact lamination of dimension p and class C 1 without holonomy is an inverse limit of compact branched manifolds of dimension p. To prove this result, we extend the triangulation theorem for C 1 manifolds to transversely Cantor C 1 laminations. In fact, we give a simple proof of this classical theorem based on the existence of C 1 -compatible differentiable structures of class C ∞ .

On fuzzy subspaces

Abstract Throughout the literature, if (X, δ) is an f.t.s., and Y ⊂ X, the induced fuzzy topological subspace (Y, δY) is defined so that δY = {ν ∧ XY: ν ∈ δ}. Since a fuzzy set isan element of Ix, it seems more reasonable to think about fuzzy topological subspaces as structures induced over each μ ∈ Ix from the fuzzy topology δ. In this sense, we introduce a new concept of fuzzy topological subspace, which coincides with the usual definition in the case that μ = χY, for Y ⊂ X. We characterize some fuzzy topological concepts, which generalize standard situations, using a certain type of prefilters.

On N-convergence of fuzzy nets

Abstract In this paper, we define the N-convergence of fuzzy nets in a fuzzy topological space and we use it to give a characterization of some fuzzy topological concepts. We give necessary conditions for the N-convergence of fuzzy nets in a fuzzy topological space and we characterize this N-convergence in a topologically generated fuzzy topological space. We also study some characterizations of fuzzy ultranets and we introduce the concept of subordination between fuzzy nets. We associate to each fuzzy net in X a prefilter on X and conversely, and we give an exhaustive description of the connections between these two theories. We give a theory of fuzzy convergence classes and we characterize fuzzy topologies via some special types of fuzzy convergence classes.

Fuzzy t -net theory

Abstract In this paper, we define the convergence of fuzzy nets in a fuzzy topological space ( X , δ ), by means of the family of its level-topologies: { l t ( δ ): t ϵ (0,1)}, introduced in [4] by Lowen. Fuzzy t -nets and fuzzy t -ultranets are also characterized and relations between t -prefilters and fuzzy t -nets are studied, getting analogous results to those of general topology.

La conjecture de Baum-Connes pour un feuilletage sans holonomie de codimension un sur une variété fermée

In [C2], Baum-Connes state a conjecture for the K-theory of C*-algebras of foliations. This conjecture has been proved by T. Natsume [N2] for C8-codimension one foliations without holonomy on a closed manifold. We propose here another proof of the conjecture for this class of foliations, more geometric and based on the existence of the Thom isomorphism, proved by A. Connes in [C3]. The advantage of this approach is that the result will be valid for all C0-foliations.

On fuzzy compactifications

Abstract The concept of a fuzzy set was introduced by Zadeh in 1965. Subsequently, Chang defined the notion of Fuzzy Topological Space in 1968. In 1980, Martin approached the problem concerning the ultracompactification of a Fuzzy Topological Space, based on the properties of the functors ω and l introduced by Lowen. This author reduces the outlined problem to that of the compactification of a Topological Space. It is the aim of this paper to give a new process of compactification for a Fuzzy Topological Space, which allows us to leave out the framework used by Martin.