On Initial and Final Characterized L- topological Groups

Department of Mathematics, Faculty of Science, El-Mansoura University, El-Mansoura, Egypt Abstract: In this research work, new topological notions are proposed and investigated. The notions are named final characterized L-spaces and initial and final characterized L-topological groups. The properties of such notions are deeply studied. We show that all the final lefts and all the final characterized L-spaces are uniquely exist in the category CRL-Sp and hence CRL-Sp is topological category over the category SET of all sets. By the notion of final characterized L-space, the notions of characterized qoutien pre L-spaces and characterized sum L-spaces are introduced and studied. The characterized L-subspaces together with their related inclusion mappings and the characterized quotient pre L-spaces together with their related canonical surjections are the equalizers and co-equalizers, respectively in CRL-Sp. Moreover, we show that the initial and final lefts and then the initial and final characterized L-topological groups uniquely exist in the category CRL-TopGrp. Hence, the category CRLTopGrp is topological category over the category Grp of all groups. By the notion of initial and final characterized L-topological groups, the notions of characterized L-subgroups, characterized product Ltopological groups and characterized L-topological quotient groups are introduced and studied., we show that the category CRL-TopGrp is concrete and co-concrete category of the category L-Top. Finally, we show that the