Ideal and some applications of simply open sets

Recently there has been some interest in the notion of a locally closed subset of a topo- logical space. In this paper, we introduce a useful characterizations of simply open sets in terms of the ideal of nowhere dense set. Also, we study a new notion of functions in topo- logical spaces known as dual simply-continuous functions and some of their fundamental properties are investigated. Finally, a new type of simply open sets is introduced

Generalized rough sets based on neighborhood systems and topological spaces

Rough sets theory is an important method for dealing with uncertainty, fuzziness and undefined objects. In this paper, we introduce a new approach for generalized rough sets based on the neighborhood systems induced by an arbitrary binary relation. Four pairs of the dual approximation operators are generated from the core of neighborhood systems. Relationship among different approximation operators are presented. We generate different topological spaces by using the core of these neighborhood systems. Relationship among different generated topologies are discussed

Computer Construction and Enumeration of All T0 and All Hyperconnected T0 Topologies on Finite Sets

There are many axioms on the principal topological spaces. Two of the interesting axioms are the T0 and hyperconnected topological spaces. There is a well-known and straightforward correspondence (cf. [2]) between the topologies on finite set Xn of n points and reflexive transitive relations (preorders) on those sets. This paper generalizes this result, characterizes the principal hyperconnected T0-topologies on a nonempty set X and gives their number on a set Xn. It mainly describes algorithms for construction and enumeration of all weaker and strictly weaker T0 and nT0-topologieson on Xn. The algorithms are written in fortran 77 and implemented on pentium II400 system

On Covering-Based Rough Intuitionistic Fuzzy Sets

Intuitionistic Fuzzy Sets (IFSs) and rough sets depending on covering are important theories for dealing with uncertainty and inexact problems. We think the neighborhood of an element is more realistic than any cluster in the processes of classification and approximation. So, we introduce intuitionistic fuzzy sets on the space of rough sets based on covering by using the concept of the neighborhood. Three models of intuitionistic fuzzy set approximation space based on covering are defined by using the concept of neighborhood. In the first and second model, we approximate IFS by rough set based on one covering (C) by defining membership and non-membership degree depending on the neighborhood. In the third mode, we approximate IFS by rough set based on family of covering (Ci) by defining membership and non-membership degree depending on the neighborhood. We employ the notion of the neighborhood to prove the definitions and the features of these models. Finlay, we give an illustrative example for the new covering rough IF approximation structure