GENERALIZED TOPOLOGICAL OPERATOR (g-Tg-OPERATOR) THEORY IN GENERALIZED TOPOLOGICAL SPACES (Tg-Spaces)\\ Part V. GENERALIZED EXTERIOR (g-Tg-EXTERIOR) AND GENERALIZED FRONTIER (g-Tg-FRONTIER) OPERATORS

In a generalized topological space Tg = (Ω,Tg) (Tg-space), g-Intg, g-Clg: P(Ω) P(Ω) (g-Tg-interior and g-Tg-closure operators) and g-Derg, g-Codg: P(Ω) P(Ω) (g-Tg-derived and g-Tg-coderived operators) are pairs of generalized topological operators which may be employed to topologize the underlying set Ω or to give characterizations of generalized operations in the generalized sense. Generalized exterior and generalized frontier operators g-Extg, g-Frg: P(Ω) P(Ω) (g-Tg-exterior and g-Tg-frontier operators), respectively, are other generalized topological operators by means of which characterizations of generalized operations under g-Intg, g-Clg: P(Ω) P(Ω) can be given without even realizing generalized interior and generalized closure operations first in order to topologize Ω in the generalized sense. In two papers, we introduced the definitions and studied the essential properties and commutativity of g-Intg, g-Clg: P(Ω) P(Ω) (g-Tg-interior} and g-Tg-closure operators) in Tg. In another two papers, we introduced the definitions and studied the essential properties of g-Derg, g-Codg: P(Ω) P(Ω) (g-Tg-derived and g-Tg-coderived operators) in Tg. Moreover, we also defined by transfinite recursion on the class of successor ordinals the δᵗʰ-iterates g-Derg(δ), g-Codg(δ): P(Ω) P(Ω) (g-Tg(δ)-derived and g-Tg(δ)-coderived operators) of g-Derg, g-Codg: P(Ω) P(Ω), respectively, and study their basic properties in a Tg. In this paper, we present novel definitions of generalized exterior and generalized frontier operators g-Extg, g-Frg: P(Ω) P(Ω), respectively, study their essential properties, and establish further characterizations of generalized operations under g-Extg, g-Frg: P(Ω) P(Ω) in Tg.