A View on Fuzzy Minimal Open Sets and Fuzzy Maximal Open Sets

A View on Fuzzy Minimal Open Sets and Fuzzy Maximal Open Sets

Hamid Reza Moradi (Islamic Azad University, Mashhad Branch, Mashhad, Iran)
DOI: 10.4018/IJSSCI.2015010104
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Abstract

A nonzero fuzzy open set () of a fuzzy topological space is said to be fuzzy minimal open (resp. fuzzy maximal open) set if any fuzzy open set which is contained (resp. contains) in is either or itself (resp. either or itself). In this note, a new class of sets called fuzzy minimal open sets and fuzzy maximal open sets in fuzzy topological spaces are introduced and studied which are subclasses of open sets. Some basic properties and characterization theorems are also to be investigated.
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1. Introduction And Basic Concepts

The notion of fuzzy set was introduced by the celebrated scientist L.A. Zadeh in the year (1965). The usefulness of the introduced notion of fuzzy set theory was realised and applied in studies in almost all branches of science and technology by many researchers in recent years.

In the year (1968), C.L. Chang introduced the concept of fuzzy topological space as an application of fuzzy sets to general topological spaces. Many researchers like R.H. Warren (1974), K.K. Azad (1981) G. Balasubramanian and P. Sundaram (1997), S.R. Malghan and S.S. Benchalli (1981) and (1984), M.N. Mukherjee and B. Ghosh (1990), A.N. Zahren (1994), J.A. Goguen (1973) and many others have contributed to the development of fuzzy topological. In the year (2001) and (2003), F. Nakaoka and N. Oda introduced and studied minimal open (resp.minimal closed) sets which are sub classes of open (resp.closed) sets. The complements of minimal open sets and maximal open sets are called maximal closed sets and minimal closed sets respectively. In the year (2000) M. Sheik john introduced and studied weakly closed sets and weakly open sets in topological spaces. Noiri (2006) introduced the concept of minimal structure on a nonempty set and also he introduced the notion of -open sets and -closed sets and characterized those sets using –cl and -int operators respectively. Also he introduced the concept of -closed sets in minimal structures which is analogous to -closed sets in a topological space introduced by Levine (1970). Further Popa and Noiri introduced -continuous functions (2007) and studied some of their basic properties. Alimohammady in (2006) introduced and extended minimal structures to fuzzy minimal structures and established some results in this setting.

In this paper, we introduce the new concepts of fuzzy minimal and fuzzy maximal sets, moreover some interesting properties and characterizations are introduced and discussed.

We recall some known definitions needed in this paper.

  • 1.1. Definition: A fuzzy subset in set is defined to be a function .

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