International Journal of Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759 www.ijmsi.org Volume 1 Issue 1 ǁ August. 2013ǁ PP-31-37 www.ijmsi.org 31 | P a g e On Nano Forms Of Weakly Open Sets M. Lellis Thivagar 1 , Carmel Richard 2 1 School of Mathematics, Madurai Kamaraj University, Madurai-625021, Tamil Nadu, India 2 Department of Mathematics, Lady Doak College, Madurai - 625 002, Tamil Nadu, India ABSTRACT : The purpose of this paper is to define and study certain weak forms of nano-open sets namely, nano  -open sets, nano semi-open sets and nano pre-open sets. Various forms of nano  -open sets and nano semi-open sets corresponding to different cases of approximations are also derived. KEYWORDS: Nanotopology,nano-open sets,nano interior, nano closure, nano  -open sets, nano semi-open sets, nano pre-open sets, nano regular open sets. 2010 AMS Subject Classification:54B05 I. INTRODUCTION Njastad [5], Levine [2] and Mashhour al et [3] respectively introduced the notions of  - open, semi-open and pre-open sets.Since then these concepts have been widelyinvestigated. It was made clear that each  -open set is semi-open and pre-open but the converse of each is not true. Njastad has shown that the family   of  - open sets is a topology on X satisfying     . The families SO(X,  ) of all semi–open sets and PO(X,  ) of all preopen sets in (X,  )are not topologies. It was proved that both SO(X,  ) and PO(X,  ) are closed under arbitrary unions but not under finite intersection. Lellis Thivagar al et [1] introduced a nano topological space with respect to a subset X of an universe which is defined in terms of lower and upper approximations of X. The elements of a nano topological space are called the nano-open sets. He has also studied nano closure and nano interior of a set. In this paper certain weak forms of nano-open sets such as nano  -open sets, nano semi-open sets and nano pre-open sets are established. Various forms of nano  - open sets and nano semi-open sets under various cases of approximations sre also derived. A brief study of nano regular open sets is also made. II. PRELIMINARIES Definition 2.1 A subset A of a space ) , (  X is called (i) semi-open [2] if )) ( ( A Int Cl A  . (ii) pre open [3] if )) ( ( A Cl Int A  . (iii)  -open [4] if ))) ( ( ( A Int Cl Int A  . (iv) regular open [4] if )) ( ( = A Cl Int A . Definition 2.2 [6] Let U be a non-empty finite set of objects called the universe and R be an equivalence relation on U named as the indiscernibility relation. Elements belonging to the same equivalence class are said to be indiscernible with one another. The pair ( ) , R U is said to be the approximation space. Let U  X . (i) The lower approximation of X with respect to R is the set of all objects, which can be for certain classified as X with respect to R and its is denoted by ) ( X L R . That is, } ) ( : ) ( { = ) ( X x R x R X L x R    U , where R(x) denotes the equivalence class determined by x. (ii) The upper approximation of X with respect to R is the set of all objects, which can be possibly classified as X with respect to R and it is denoted by ) ( X U R . That is, ) ( X U R = } ) ( : ) ( {     X x R x R x  U (iii) The boundary region of X with respect to R is the set of all objects, which can be classified neither as X nor