Semi-Generalized Closed Set in the Closure Spaces

. In this paper, we introduce the concept of semi generalized closed set (= 𝒮𝒢 − closed) sets, and semi generalized open (= 𝒮𝒢 − open) sets in the closure space .Furthermore, we study some of their properties. And investigate the relation between them .

The purpose of this paper is to introduce and study the concept of semi generalized closed sets in closure spaces .Closure spaces were introduced by E.Čech [1] in 1966 and then studied by many mathematicians, see e.g.[2], [3], [4] and [8].Closure spaces are sets endowed with a grounded, extensive and monotone closure operator.Khampakdee [6],defined an study g-closed sets (2008), and then in [5], (2009)  As a continuation of this work, we introduce and study in Section 3, a new class of sets namely − closed sets which is properly placed in between the class of semi-closed sets and the class of g-closed sets .In Section 4, the class of − open sets introduced and investigated.All definitions of the several concepts used throughout the sequel are explicitly stated in the following section.

preliminaries
Definition 2.1.[1] Let :P(M) →P(M) be a function identified on a power set P(M) of the set M,  will be the closure operator over M and the couple (M, ) is called the closure space, if the following axioms are satisfied : Definition 2.2.[1] let  be the closure operator on ℳ named idempotent when A⊆ M then   (A) =  (A), and we say  is called additive if , ℬ are subset of ℳ then ()∪(ℬ)=(∪ℬ).Definition 2.3.A function int :P(M) →P(M) identified on the power set P(M) from the set M, the interior operator on M is called an interior operator that satisfies: (1) int(M) = M, (2) int(A) ⊆ A, for each A,B ⊆ M.
Similarly, given a set ℳ with an interior operator i, we identified an operator i : P(ℳ) → P(ℳ) by i () = ℳ − i(ℳ − ).Definition 2.5.[5] A subset A of ℳ is said to be closed over the closure space (ℳ, ) when  (A) = A. In addition, subset B of ℳ is named open when its complement (ℳ\B) is closed.The empty set and the entire space where both open and closed Simultaneously.Definition 2.6.[5] The closure space (N, E) named is subspace of (ℳ, Definition 2.8.[6] The subset Q of closure space (ℳ, )which named is generalized closed set in the closure space (g-closed) if CQ⊆G whenever Q⊆G and G is open set over M. A subset A ⊑ℳ is called a generalized -open (g-open) set if its complement is generalized closed set.Definition 2.9.A subset A of a closure space (ℳ, ) is said to be : (1) a semi-open set [5]

semi generalized closed set in the closure space
This section is dedicated to the introduction and discussion of the basic properties of the notion of a semi-generalized closed set.Definition 3.1.[5] The subset Q of closure space (ℳ, ) which named is semi-generalized closed set in the closure space introduced the notion of semi open sets in closure spaces and showed their fundamental properties.The semi-open sets are used to define semi-open maps.

( 2 )
a clopen set if A is open and closed set at the same time.Lemma 2.10.Let A be a subset of a closure space (ℳ, ).Then: (1) the intersection of two g-open set is g-open set .(2) if A is semi-closed set then A⊑ i((A)).(3) if A⊑B⊑ℳ and A is g-open in B , B is g-open in ℳ then A is g-open in ℳ. Proposition 2.11.let (M, ) be the closure space and ⊆ .Where a subset Q is open if and only if ⊆  Anyplace is closed and ⊆ Q.