Giới thiệu Một lớp học tổng quát của bộ khép kín đã được xem xét bởi Maki trong năm 1986 [9]. Ông điều tra các bộ có thể được biểu diễn như là công đoàn của bộ khép kín và được gọi là V-bộ. | Vietnam Journal of Mathematics 34 Viet n a m J o u r n a I of MATHEMATICS VAST 2006 Strong Insertion of a Contra - Continuous Function Majid Mirmiran Department of Mathematics University of Isfahan Isfahan 81746-73441 Iran Abstract. . 1. Introduction . . . . contra- continuous Thls work was supported by University of Isfahan . 821033 and Centre of Excellence for Mathematics University of Isfahan . Majid Mirmiran . .property. . . . .weak insertion property for A . . .strong insertion property for A . . . . 2. The Main Results Definition . Let A be a subset of a topological space. We define the subsets A and A V as follows A .and A A . . .kernel A . Definition . If A is a binary relation in a set A then is defined as follows A A if and only if A A A implies A A A and A A A implies A A A for any A and A in A . Definition . A binary relation A in the power set A A of a topological space is called a strong binary relation in A A in case A satisfies each of the following conditions A If A i A j for any BA. and for any BA. then there exists a set A in A A such that A i BA and A A A j for any. and any. A If A A then A A . A If A BA then A A and A A V. Strong Insertion of a Contra - Continuous Function Definition . If is a real-valued function defined on a space and if .for a real number then .is called a lower indefinite cut set in the domain of at the level . Theorem . Let and be real-valued functions on a topological space in which sets are open with .If there exists a strong binary relation on the power set of and if there exist lower indefinite cut . in the domain of and at the level for each rational number such that if . t then. then there exists a contra-continuous function defined on such that. Proof. . . . .V . H . .- . . . . . . . . Theorem . Let . and . be properties and be a space satisfying the weak insertion property for . . Also assume that and are functions on such property . and has property . The space has the .
