Studies on Multiset Topologies and its Generalizations

Abstract

Many generalizations of the set theory have been developed to overcome the natural newlinedifficulties that arise in real-life problems that cannot be solved by the classical newlinetwo-valued logic and set theory introduced by Georg Cantor. The first one among newlinethem is the fuzzy set theory introduced by L. A. Zadeh in 1965. Unlike in the case of newlineclassical set theory, an element can take any membership value between 0 and 1 in newlinefuzzy set theory and classical set theory has become a special case of fuzzy set with newlineonly two possible membership values 0 and 1. The concept of the fuzzy set has given newlinea general framework and it has become a useful mathematical tool to solve several newlineproblems relating to vagueness in various fields. newlineAfter this first generalization of classical set theory, many generalizations were newlineintroduced. Rough sets by Pawlak, Multisets by Yager, Soft sets by Moldostov, Genuine newlinesets by Demicri are some of the other alternatives. All have many applications newlineand are used to solve several real-life problems. newlineOne among these generalizations is the concept of the multisets (msets). Just like newlinefuzzy sets, multisets also have a membership function, but it represents the number newlineof times an element repeats in the set. In the case of multisets, it is called multiplicity newlinefunction and it maps each member of the underlying set to a non-negative integer that newlinegives how many times it occurs in the multiset. Hence a multiset can be considered newlineas a set with repeated elements. The occurrence of multisets is quite natural since newlinethere are many identical things in nature like repeated hydrogen atoms in a water newlinemolecule, repeated observations in statistical data, repeated zeros of a polynomial, newlineetc.Many set-theoretic structures and algebraic structures were already developed in newlinemultiset theory. The concept of multiset topology ( M-topology ) was also introduced.Many of the M-topological concepts like M-compactness, M-connectedness, semicompactness, newlinegeneralized closed sets, rb-closed sets, and rb-convergence of multisets newlinealso were developed.