R. Lowen

Fuzzy topological spaces and fuzzy compactness

Abstract It is the purpose of this paper to go somewhat deeper into the structure of fuzzy topological spaces. In doing so we found we had to alter the definition of a fuzzy topology used up to now. We shall also introduce two functors \ gw and \ gi which will allow us to see more clearly the connection between fuzzy topological spaces and topological spaces. Finally we shall introduce the concept of fuzzy compactness as the generalization of compactness in topology. It will be shown in a following publication that contrary to the results obtained up to now, the Tychonoff-product theorem is safeguarded with fuzzy compactness.

A comparison of different compactness notions in fuzzy topological spaces

In this paper we study the different kinds of compactness notions that have been introduced up to this time. We restrict ourselves to fuzzy topological spaces as defined in [4]. In Section 1 we give some alternative characterizations. For a good definition of fuzzy compactness we will demand that in the special case of topological spaces it coincides with the usual notion of compactness. In Section 2 we show which compactness notions are good extensions and which are not. Moreover, we want the fundamental property of compactness in topological spaces, namely, the Tychonoff theorem on products, to be fulfilled in the more general setting of fuzzy topological spaces. In Section 3 we see for which notions there is a .product theorem. In Section 4 we study the implications that exist between the different notions, and in Section 5 we give some concluding remarks. 1.

Fuzzy neighborhood spaces

Abstract In this paper we introduce a subcategory of the category of fuzzy topological spaces and we prove basic properties of this subcategory. The main idea behind these spaces is the notion of fuzzy neighborhoods which enables us to introduce a local theory of fuzzy topologies.

Fuzzy Uniform Spaces

In this paper we introduce the notion of fuzzy uniform spaces. In Section 2 we give some basic results and show how a fuzzy topology is derived from a fuzzy uniformity. In Sections 3 and 4 we prove that the notions which are introduced are good extensions and we prove that the category of uniform spaces is nicely injected in the category of fuzzy uniform spaces. Finally in Section 5 we prove some basic results

Convex fuzzy sets

Abstract This paper gathers some elementary known results about convex fuzzy sets and completes the theory, introducing the necessary concepts. Using a representation theorem for fuzzy subspaces it gives separation theorems for convex fuzzy sets in the proper setting.

Initial and Final Fuzzy Topologies and the Fuzzy Tychonoff Theorem

In this paper we introduce the notions of initial and final fuzzy topologies. In Theorems I .4 and 2.4, we show that from a categorical point of view they are the right concepts to generalize the topological ones. In Theorem 1.6, we show that with our notion of fuzzy compactness the Tychonoff product theorem is safe- guarded. We also show that this is not the case for weak fuzzy compactness. For notions and results used but not defined or shown in this paper, we refer the reader to [lo]. 1.

FUZZY PROBABILITY MEASURES

Abstract In [4] Hohle has defined fuzzy measures on G-fuzzy sets [2] where G stands for a regular Boolean algebra. Consequently, since the unit interval is not complemented, fuzzy sets in the sense of Zadeh [8] do not fit in this framework in a straightforward manner. It is the purpose of this paper to continue the work started in [5] which deals with [0,1]-fuzzy sets and to give a natural definition of a fuzzy probability measure on a fuzzy measurable space [5]. We give necessary and sufficient conditions for such a measure to be a classical integral as in [9] in the case the space is generated. A counterexample in the general case is also presented. Finally it is shown that a fuzzy probability measure is always an integral (if the space is generated) if we replace the operations ∧ and ∨ by the t-norm To and its dual S0 (see [6]).

The categorical topology approach to fuzzy topology and fuzzy convergence

It is the purpose of this paper to have a look at fuzzy topology from the point of view of categorical topology. We first look at interesting subcategories of FTS and determine whether they are bireflective and/or coreflective. We also describe the set of all subcategories of FTS which are at the same time bireflective and coreflective. Due to the fact that TOP is one of these subcategories of FTS, FFS itself can be neither Cartesian closed nor extensional. We therefore, secondly, construct a supercategory of bTS which is a topological quasitopos. The construction of this quasitopos is based on the theory of convergence in FTS.

Handbook of the History of General Topology

Introduction. Combinatorial Topology Versus Point-set Topology I.M. James. Elements of the History of Locale Theory P. Johnstone. Nonsymmetric Distances and their Associated Topologies: About the Origins of Basic Ideas in the Area of Asymmetric Topology H.-P.A. Kunzi. Supercategories of Top and the Inevitable Emergence of Topological Constructs E. Lowen-Colebunders, R. Lowen. Topological Features of Topological Groups M.G. Tkachenko. History of Shape Theory and its Application to General Topology S. Mardesic, J. Segal. A History of the Normal Moore Space Problem P.J. Nyikos. Index.

A fuzzy lagrange interpolation theorem

Abstract The following problem was posed by L.A. Zadeh: “Suppose we are given n + 1 points x0, …, xn in R , and for each of these points a ‘fuzzy value’ in R , rather than a crisp one. Is it then possible to construct some function on R with range also a collection of ‘fuzzy values’; which coincides, on the given n + a points, with the given ‘fuzzy values’; and which fulfills some natural ‘smoothness’ condition?” In this paper we shall present a solution to this problem, based on the fundamental and well-known polynomial interpolation theorm of Lagrange.