# A note on compactness in a fuzzy setting

TL;DR: Fuzzy filter bases are used to introduce the notion of Compactness in fuzzy setting and the expected basic properties of compactness are explored.

Abstract: The concept of filter basis in a fuzzy setting is defined and investigated. Fuzzy filter bases are then used to introduce the notion of compactness in fuzzy setting. The expected basic properties of compactness are explored.

Topics: Fuzzy set operations (67%), Fuzzy number (65%), Fuzzy classification (64%), Defuzzification (63%), Fuzzy mathematics (62%)

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TL;DR: Almost compact fuzzy sets are characterized in terms of the concept of fuzzy filterbase in order to establish properties of such sets using the concepts of fuzzy ds-closure and fuzzyDs-cluster points of fuzzyfilterbase.

Abstract: The study of almost compact fuzzy topological spaces was initiated by Concilio and Gerla [4], which was followed by further investigations of the same in [11]. The present paper aspires for an extension of the same concept to fuzzy sets in a fuzzy topological space. We characterize almost compact fuzzy sets in terms of the concept of fuzzy filterbase. Some properties of such sets are established using the concepts of fuzzy ds-closure and fuzzy ds-cluster points of fuzzy filterbase.

15 citations

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TL;DR: The concept of a fuzzy almost c-continuity is introduced and some of its properties are investigated and it is shown that the properties of this type of consistency are similar to those of discrete-time consistency.

Abstract: In this paper we introduce the concept of a fuzzy almost c-continuity and investigate some of its properties.

10 citations

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TL;DR: The concept of fuzzy compact-open topology is introduced and some characterizations of this topology are discussed.

Abstract: The concept of fuzzy compact-open topology is introduced and some characterizations of this topology are discussed.

6 citations

### Cites background from "A note on compactness in a fuzzy se..."

...2 of [6], it follows that 17 is fuzzy compact....

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...13 of [6], U is fuzzy closed; thus U = 0....

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...Since X is fuzzy Hausdorff (a fuzzy compact subspace of a fuzzy Hausdorff space is fuzzy closed [6]) U is fuzzy closed: thus U = 0....

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TL;DR: The relations among various types of continuity of fuzzy proper function on a fuzzy set and at fuzzy point belonging to the fuzzy set in the context of Sostak's I-fuzzy topological spaces are discussed.

Abstract: The relations among various types of continuity of fuzzy proper function on a fuzzy set and at fuzzy point belonging to the fuzzy set in the context of Sostak's I-fuzzy topological spaces are discussed. The projection maps are dened as fuzzy proper functions and their properties are proved.

4 citations

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Abstract: The investigation of the fuzziness of a non-equilibrium thermodynamic system is justified by the structural and parametric uncertainties of such systems. The paper gives a fuzzy set formulation of the phenomenological equations and shows a realistic approach for studying the entropy production in physical systems, the time trajectories of chemical reactions, etc. Using algorithms derived for special reaction systems, bundles of time trajectories with prescribed boundary possibility measures are calculated.

4 citations

##### References

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TL;DR: This paper explores the foundations of, generalizes, and continues the work of Zadeh in [I] and [2].

2,222 citations

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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.

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.

855 citations

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Abstract: 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.

263 citations