A unified theory of weakly g-closed sets and weakly g-continuous functions

Takashi Noira, Valeriu Popa
2013 Sarajevo Journal of Mathematics  
We introduce the notion of weakly mng-closed sets as a unified form of weakly ω-closed sets [38] , weakly rg-closed sets [23], weakly πg-closed sets [40] and weakly mg * -closed sets [29] . Moreover, we introduce and study the notion of weakly mng-continuous functions to unify some modifications of weakly g-continuous functions. 2010 Mathematics Subject Classification. 54A05, 54C08. 130 TAKASHI NOIRI AND VALERIU POPA g-continuous functions. By using m-continuity, we obtain several
more » ... veral characterizations and properties of weakly mng-continuous functions. Preliminaries Let (X, τ ) be a topological space and A a subset of X. The closure of A and the interior of A are denoted by Cl(A) and Int(A), respectively. A subset A is said to be regular open if A = Int(Cl(A)). We recall some generalized open sets in a topological space. π-open [45] if A is the finite union of regular open sets. The family of all α-open (resp. semi-open, preopen, b-open, β-open, πopen, regular open) sets in (X, τ ) is denoted by α(X) (resp. SO(X), PO(X), BO(X), β(X), π(X), RO(X)). preopen, b-open, β-open, π-open). Definition 2.3. Let (X, τ ) be a topological space and A a subset of X. The intersection of all α-closed (resp. semi-closed, preclosed, b-closed, β-closed, π-closed) sets of X containing A is called the α-closure [19] (resp. semiclosure [11], preclosure [14], b-closure [4], β-closure [2], π-closure) of A and is denoted by αCl(A) (resp. sCl(A), pCl(A), bCl(A), β Cl(A), πCl(A)). Definition 2.2. Let (X, τ ) be a topological space. A subset Definition 2.4. Let (X, τ ) be a topological space and A a subset of X. The union of all α-open (resp. semi-open, preopen, b-open, β-open, π-open) sets of X contained in A is called the α-interior [19] (resp. semi-interior [11], preinterior [14], b-interior [4], β-interior [2], π-interior) of A and is denoted by αInt(A) (resp. sInt(A), pInt(A), bInt(A), β Int(A), πInt(A)). Minimal structures and m-continuity Definition 3.1. Let X be a nonempty set and P(X) the power set of X. A subfamily m X of P(X) is called a minimal structure (briefly m-structure) on X [32], [33] if ∅ ∈ m X and X ∈ m X .
doi:10.5644/sjm.09.1.12 fatcat:h7czrf5zfrhrbiukw46ohkhgoq