Ratna Dev Sarma

Fuzzy nets and their application

Fuzzy net-theory is enriched by the introduction of the fuzzy net of fuzzy sets. The limsup, liminf and limit of a fuzzy net of fuzzy sets are defined and their various properties are discussed. Alternative characterizations based on the notion of a fuzzy net are provided for several fuzzy topological concepts including open and closed fuzzy sets, fuzzy continuity, maps with closed fuzzy graph, open fuzzy mapping, etc. Thus the net theoretic approach is shown to be a promising tool in fuzzy topology.

Various fuzzy topological notions and fuzzy almost continuity

Abstract We extend the notion of almost continuity in the sense of Husain (1966) to the fuzzy setting. We obtain a pointwise characterization of the notion by dual point and fuzzy net. Also we formulate conditions under which the two notions of fuzzy almost continuity - the one defined in this paper and the one defined by Ajmal and Azad (1990) - turn out to be equivalent. The main idea behind this work is to show that using the localized approach developed in earlier papers, the study of fuzzy almost continuity can be carried out as successfully as that of its counterpart in general topology: In the process, we exhibit the compatibility of various existing fuzzy topological notions such as fuzzy regularity, c i -connectedness ( i = 3, 4), c i -local connectedness ( i = 3, 4), fuzzy T 1 -axiom, fuzzy T 2 -axiom, etc.

Fuzzy subcontinuity, inverse fuzzy subcontinuity and a new category of fuzzy topological spaces

The notions of subcontinuity and inverse subcontinuity are extended to the fuzzy setting. Adopting fuzzy net-theoretic approach, various properties of fuzzy subcontinuous and inversely fuzzy subcontinuous mappings are studied. A category of fuzzy topological spaces is constructed. Canonical example is constructed by using fuzzy translation of fuzzy numbers introduced by Rodabaugh to show that fuzzy subcontinuity does not imply kfuzzy continuity. In the process, N-compactness is characterized and fuzzy unit interval is shown to be N-compact. Also a direct proof is provided showing that fuzzy addition of fuzzy numbers is fuzzy continuous.

Category N and functions with closed fuzzy graph

Abstract A net-theoretic approach for defining a fuzzy topological space is provided. The fuzzy topological spaces so obtained form a category which we denote by N in this paper. It is found that the limitations exhibited by the Chang category particularly with respect to projections, closure of a fuzzy set, compactness etc. no longer exist in N. The notion of closed fuzzy graph is investigated. The divergences shown by this notion are immediately put to rest if the study is carried out in N, instead of C , the Chang category.

A new approach to fuzzy topological spaces and various fuzzy topological notions defined by nets

N-fuzzy spaces, introduced by the authors, are further investigated. Various fuzzy topological motions such as fuzzy local compactness, functions with closed fuzzy graph, fuzzy subcontinuity, inverse fuzzy subcontinuity, etc., are studied in the light of N(I). These notions are found to be more coherent in N(I) than in the Changs category.

Regular Sets in Topology and Generalized Topology

We define and study a new class of regular sets called -regular sets. Properties of these sets are investigated for topological spaces and generalized topological spaces. Decompositions of regular open sets and regular closed sets are provided using -regular sets. Semiconnectedness is characterized by using -regular sets. -continuity and almost -continuity are introduced and investigated.

A new approach to fuzzy topological spaces and fuzzy perfect mappings

Abstract 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 N here, are discussed. The notion of a perfect mapping is introduced and studied in the fuzzy setting. It is found that in N , a fts is N-compact iff any mapping ƒ: X → {x}, where {x} is any singleton, is fuzzy perfect. A fuzzy continuous mapping ƒ: X → Y from an N-compact fts X to a Hausdorff fts Y is fuzzy perfect in N . However, these results do not hold in C , the Changs category. In general, it is found that the divergences in the study of fuzzy perfect mappings disappear if the study is carried out in N , instead of C.