Mθ - Closed Set In T0 -MAlexandroff Spaces

2019 University of Thi-Qar journal  
In this paper, we introduce definitions of Mθ-closed, Mθ-adherent and Mθ-open set, and we give some theorems and results related to these sets in minimal Alexandroff space. Web Site: https://jutq.utq.edu.iq/index.php/main Email: journal@jutq.utq.edu.iq 5 the minimal θ-adherent of S and is denoted by Mθ-Cl(S). Definition 3.8 Let (X,M) be an minimal space has the property I , S  X is called minimal θ-closed if Mθ-Cl(S) = S , and denoted by Mθ-closed .The complement of an Mθ-closed set is called
more » ... osed set is called minimal θ-open (Mθ-open). Definition 3.9 A point x ∊X is said to be an Mθ -interior point of A if there exists an mopen set U containing x such that U  m-cl(U) A. The set of all Mθinterior points of A is said to be the Mθ -interior of A , and it is denoted by Remark 3.10 Let (X, M) be a minimal space has the property I. (iii) Every Mθ-open set is m-open. Theorem 3.11 If X is aT 0 -M A -Space and A is am-open subset of X, then Mθ.Cl(A)= m-Cl(A).Proof . ⇒m-Cl (A) Mθ-Cl (A) , it is clear from Remark (3.10) . Let x Mθ-Cl(A) then for each M-open set U containing x ,then(m-Cl(U) ⋂ A) Ø. So there exists a A such that am-Cl (U). Therefore for some vU , and hence v  a ⋂U, since a  A, we have A ⋂ U Ø, therefore xm-Cl (A). Theorem 3.12 Let X be a T 0 -M A -Space and let A be a non-empty subset of X then m-Cl ( A) = Mθ-Cl (A). .iq 6 Proof. Suppose that x Mθ-Cl (A) and V be an m-open set containing x then m-Cl(V)⋂AØ .So there exists a m-Cl(V) and a A. Thus a  u for some u V, therefore u , hence u A ⋂ V  Ø . Hence x m-Cl ( A). Corollary 3.13 Let X be a T 0 -M A -Space, and let A be a non-empty m-open set of X. Then x  Mθ-Cl (A) ,for every xMθ-Cl (A) . Proof. Because A is an m-open set , then A= A , hence m-Cl (A) = m-Cl ( A) =Mθ-Cl (A) is an m-closed set of m-space X , thus x Mθ-Cl (A). Corollary 3.14 Let X be a T 0 -M A -Space, and let A be a non-empty set of X , then m-Cl( A) = Mθ-Cl( A). Proof. Since A  M . So by the theorem (3.12) , m-Cl ( A) = Mθ-Cl ( A) Theorem 3.15 Let X be aT 0 -M A -Space, and let A be a nonempty m-open set of X, then x⋂Mθ-Cl(A) = Ø  xMθ-Cl(A). Proof. Suppose to contrary that , there exists x Mθ-Cl (A) such that x⋂ Mθ-Cl(A))  Ø. So , there exists u  x and uMθ-Cl(A) , since from corollary (3.13) we get x u  Mθ-Cl(A)which is a contradiction . Theorem 3.16 Let X be a T 0 -M A -Space, and let A be a nonempty m-open set of X. Then A is Mθ-open set if and only if A is Mθ-closed. Proof. A is Mθ-open iff ⩝x A,m-Cl(↱x) A iffm-Cl(↱A)  A. But A  m-Cl(↱A) so A is Mθ-open iff m-Cl(↱A) = A = Mθ-Cl( A). From Theorem
doi:10.32792/utq/utj/vol12/2/1 fatcat:6awqyfsoonhsjentjtfpvztzbq