A Study of Strongly Generalized Compact Spaces with Applications to SGCGC-Spaces and Coercive Mappings

This paper introduces the notions of strongly generalized compact (sg-compact) spaces, sgcgc-sets and sgcgc-spaces, and strongly generalized coercive (sg-coercive) functions as natural strengthenings of the g-compact spaces, cgc-spaces, gcgc-spaces, and g-coercive functions introduced by Al-Janabi and Johnny. Building on the foundational theories of generalized closed sets (Levine), g-compactness (Selvarani; Caldas, Jafari, Moshokoa and Noiri), and generalized continuity (Balachandran, Sundaram and Maki), we establish that every sg-compact space is g-compact, and provide characterizations of sg-compactness via nets and the finite intersection property in g*-spaces. New preservation theorems are proved: the sg-irresolute continuous image of an sg-compact space is sg-compact, and the sg-irresolute inverse image of an sgcgc-set is sgcgc. We further prove that the composition of two sg-coercive functions is sg-coercive, and that on g*-Hausdorff spaces g-coercive gI-continuous functions are precisely the gI-compact ones. A Tychonoff-type theorem asserts that a finite product of sg-compact g*-spaces is sg-compact. Several examples and counterexamples illustrate that the implications among compactness notions are, in general, strict.