The Operators' Theorems on Fuzzy Topological Spaces

2018

An idea of fuzzy norm on a linear space introduced by Katsaras [11] in 1984. He studied fuzzy topological vector spaces. Following his pioneering work, Felbin [6] offered in 1992 an alternative definition of a fuzzy norm linear space (FNLS). With this definition of fuzzy normed linear space, it has been possible to introduce a notion of fuzzy bounded linear operator over fuzzy normed linear spaces to define “fuzzy norm” for such an operator. In [6], Felbin introduced an idea of fuzzy bounded operators and defined a fuzzy norm for such an operator which was erroneous as it shown in Example 3.1 of [3]. Xiao and Zhu ([14], [15]) studied various properties of Felbin-type fuzzy normed linear spaces. They gave a new definition for the norm of the bounded operators. A different definition of a fuzzy bounded linear operator and a “fuzzy norm” for such an operator was introduced by Bag and Samanta [3]. Finally, we note that the definition of the fuzzy norm of an operator was given in [3] and...

Advances in Pure Mathematics, 2021

Topology has enormous applications on fuzzy set. An attention can be brought to the mathematicians about these topological applications on fuzzy set by this article. In this research, first we have classified the fuzzy sets and topological spaces, and then we have made relation between elements of them. For expediency, with mathematical view few basic definitions about crisp set and fuzzy set have been recalled. Then we have discussed about topological spaces. Finally, in the last section, the fuzzy topological spaces which is our main object we have developed the relation between fuzzy sets and topological spaces. Moreover, this article has been concluded with the examination of some of its properties and certain relationships among the closure of these spaces.

Journal of Intelligent & Fuzzy Systems, 2017

In classical functional analysis, continuous mappings are essential for the study of many important theorems such as open mapping theorem, closed graph theorem and uniform boundedness theorem. These results are yet to be established in the general setting of an intuitionistic fuzzy n-normed linear space (IFnNLS). This motivates us to introduce the notion of continuous linear operators in this generalized setting. Furthermore, we establish the uniform continuity theorem and Banach's contraction principle in an IFnNLS.

Iranian Journal of Fuzzy Systems, 2011

In this note, we aim to present some properties of the space of all weakly fuzzy bounded linear operators, with the Bag and Samanta's operator norm on Felbin's-type fuzzy normed spaces. In particular, the completeness of this space is studied. By some counterexamples, it is shown that the inverse mapping theorem and the Banach-Steinhaus's theorem, are not valid for this fuzzy setting. Also finite dimensional normed fuzzy spaces are considered briefly. Next, a Hahn-Banach theorem for weakly fuzzy bounded linear functional with some of its applications are established.

International Journal of Fuzzy Systems and Advanced Applications, 2021

The fuzzy topological space was introduced by Dip in 1999 depending on the notion of fuzzy spaces. Dip’s approach helps to rectify the deviation in some definitions of fuzzy subsets in fuzzy topological spaces. In this paper, further definitions, and theorems on fuzzy topological space fill the lack in Dip’s article. Different types of fuzzy topological space on fuzzy space are presented such as co-finite, co-countable, right and left ray, and usual fuzzy topology. Furthermore, boundary, exterior, and isolated points of fuzzy sets are investigated and illustrated based on fuzzy spaces. Finally, separation axioms are studied on fuzzy spaces

2012

We introduce the concepts of generalized fuzzy topological spaces, generalized fuzzy neighborhood systems, in Sostak sense. Furthermore, we construct generalized fuzzy interior, generalized fuzzy closure on generalized fuzzy topological space and generalized fuzzy neighborhood system. Also, we introduce the concepts of generalized fuzzy () , y y ¢-continuous. We study their properties and discuss the relationships between these concepts.

Fuzzy Sets and Systems, 1996

A net-theoretic approach is provided for defining a fuzzy topological space. Fuzzy topological spaces so defined form a category. The main features and advantages of this category, which is denoted by ..,V here, are discussed. The notion of a perfect mapping is introduced and studied in the fuzzy setting. It is found that in ./V', a Its is N-compact iff any mapping f: X-, Ix}, where {x} is any singleton, is fuzzy perfect. A fuzzy continuous mapping f: X ~ Y from an N-compact Its X to a Hausdorff Its Y is fuzzy perfect in JV'. However, these results do not hold in C, the Chang's category. In general, it is found that the divergences in the study of fuzzy perfect mappings disappear if the study is carried out in ,l , instead of C.

The concept of dislocated fuzzy metric space is introduced and the associated fuzzy topologies are discussed. Generalized fuzzy versions of the Banach contraction mapping theorem is also proved.

IRAQI JOURNAL OF SCIENCE

The principal aim of this research is to use the definition of fuzzy normed space to define fuzzy bounded operator as an introduction to define the fuzzy norm of a fuzzy bounded linear operator then we proved that the fuzzy normed space FB(X,Y) consisting of all fuzzy bounded linear operators from a fuzzy norm space X into a fuzzy norm space Y is fuzzy complete if Y is fuzzy complete. Also we introduce different types of fuzzy convergence of operators.

Rocky Mountain Journal of Mathematics, 1978

It is shown that fuzzy continuous functions can be characterized by the closure of fuzzy sets, a subbasis for a fuzzy topology and fuzzy neighborhoods. Additional results are obtained concerning the collection of all fuzzy topologies on a fixed set, the interior of a fuzzy set, the closure of a fuzzy set, a fuzzy limit point, the derived fuzzy set and the relative fuzzy topology.