DEMONSTRATIO MATHEMATICA Vol. XLV No 4 2012 C. Carpintero, N. Rajesh, E. Rosas m-PREOPEN SETS IN BIMINIMAL SPACES Abstract. The aim of this paper is to introduce and characterize the concepts of preopen sets and their related notions in biminimal spaces. 1. Introduction In [4], Popa and Noiri introduced the notion of minimal structure which is a generalization of a topology on a given nonempty set. They also introduced the notion of m-continuous function as a function defined between a mini- mal structure and a topological space. They showed that the m-continuous functions have properties similar to those of continuous functions between topological spaces. Let X be a topological space and A ⊂ X . The closure of A and the interior of A are denoted by Cl(A) and Int(A), respectively. A subfamily m of the power set P (X ) of a nonempty set X is called a mini- mal structure [4] on X if ∅ and X belong to m. By (X, m), we denote a nonempty set X with a minimal structure m on X . The members of the minimal structure m are called m-open sets [4], and the pair (X, m) is called an m-space. The complement of m-open set is said to be m-closed [4]. In this paper we introduce and characterize the concepts of preopen sets in a biminimal space (X, m 1 ,m 2 ), which is a set X with two arbitrary minimal structures m 1 and m 2 . 2. Preliminaries In this section, we recall the m-structure and the m-operator notions. Also, we recall some important subsets associated to these concepts. Definition 2.1. [1] Let X be a nonempty set and let m X ⊆ P (X ), where P (X ) denote the set of power of X . We say that m X is an m-structure (or a minimal structure) on X , if ∅ and X belong to m X . 2000 Mathematics Subject Classification : 54D10. Key words and phrases : biminimal spaces, (i, j )-m-preopen sets, (i, j )-m-preclosed sets. 10.1515/dema-2013-0414