Fuzzy points and local properties of fuzzy topology
01 May 1974-Journal of Mathematical Analysis and Applications (Academic Press)-Vol. 46, Iss: 2, pp 316-328
About: This article is published in Journal of Mathematical Analysis and Applications.The article was published on 1974-05-01 and is currently open access. It has received 251 citations till now. The article focuses on the topics: Fuzzy set operations & Fuzzy classification.
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995 citations
TL;DR: 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.
894 citations
01 Aug 1996
TL;DR: The calculus of fuzzy restrictions is concerned with translation of propositions of various types into relational assignment equations, and the study of transformations of fuzzy Restrictions which are induced by linguistic modifiers, truth-functional modifiers, compositions, projections and other operations.
Abstract: A fuzzy restriction may be visualized as an elastic constraint on the values that may be assigned to a variable In terms of such restrictions, the meaning of a proposition of the form “x is P,” where x is the name of an object and P is a fuzzy set, may be expressed as a relational assignment equation of the form R(A(x)) = P, where A(x) is an implied attribute of x, R is a fuzzy restriction on x, and P is the unary fuzzy relation which is assigned to R For example, “Stella is young ,” where young is a fuzzy subset of the real line, translates into R(Age(Stella))= young The calculus of fuzzy restrictions is concerned, in the main, with (a) translation of propositions of various types into relational assignment equations, and (b) the study of transformations of fuzzy restrictions which are induced by linguistic modifiers, truth-functional modifiers, compositions, projections and other operations An important application of the calculus of fuzzy restrictions relates to what might be called approximate reasoning , that is, a type of reasoning which is neither very exact nor very inexact The main ideas behind this application are outlined and illustrated by examples
579 citations
Cites background or methods from "Fuzzy points and local properties o..."
...2A more detailed discussion of this and related issues may be found in [3], [4] and [5]....
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...In the development of a parallel theory based on fuzzy sets, many interesting phenomena have been observed [3-6]....
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...cept of point can also be fuzzfied and a local theory is therefore possible [5]....
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...Fuzzy algorithms and fuzzy programs were introduced in [2,4,5,6,10]....
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...tion "fuzzy measure" and a functional "fuzzy integral" in the former papers [4, 5]....
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01 Jan 2001
TL;DR: In mathematics, certain notions of topology are also abstractions of classical concepts in the study of real or complex functions, including open sets, continuity, connectedness, compactness, and metric spaces.
Abstract: Topology has its roots in geometry and analysis. From a geometric point of view, topology was the study of properties preserved by a certain group of transformations, namely the homeomorphisms. Certain notions of topology are also abstractions of classical concepts in the study of real or complex functions. These concepts include open sets, continuity, connectedness, compactness, and metric spaces. They were a basic part of analysis before being generalized in topology.
473 citations
01 Jan 1999
TL;DR: This paper gives the first comprehensive account on various systems of axioms of fixed-basis, L-fuzzy topological spaces and their corresponding convergence theory.
Abstract: This paper gives the first comprehensive account on various systems of axioms of fixed-basis, L-fuzzy topological spaces and their corresponding convergence theory. In general we do not pursue the historical development, but it is our primary aim to present the state of the art of this field. We focus on the following problems:
394 citations
References
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1,997 citations
01 Jan 1972
TL;DR: A basic idea suggested in this paper is that a linguistic hedge such as very, more or less, much, essentially, etc. may be viewed as an operator which acts on the fuzzy set representing the meaning of its operand.
Abstract: A basic idea suggested in this paper is that a linguistic hedge such as very, more or less, much, essentially. slightly, etc. may be viewed as an operator which acts on the fuzzy set representing the meaning of its operand. For example, in the case of the composite term very tall man, the operator very acts on the fuzzy meaning of the term tall man. To represent a hedge as an operator, it is convenient to define several elementary operations on fuzzy sets from which more complicated operations may be built up by combination or composition. In this way, an approximate representation for a hedge can be expressed in terms of such operations as complementation, intersection, concentration, dilation, contrast intensification, fuzzification, accentuation, etc. Two categories of hedges are considered. In the case of hedges of Type I, e.g., very, much, more or less, slightly, etc., the hedge can be approximated by an operator acting on a single fuzzy set. In the case of hedges of Type II, e.g., technical...
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