A New Study on Quasi 2-Normed Spaces: Geometric, Topological and Metrizable

This research paper addresses several aspects of the structure of quasi- -normed spaces, which dates back to C. Park, as a broader class of quasi-modular spaces. It begins by reviewing the definition along with several diverse examples. Also, the derivation of a normed space, a quasi-normed space, and a quasi-2-metric space is presented from the current space and vice versa. The description of the ball in this space was presented as a non-convex, symmetric, balanced, and absorbing set, and this was proven. Topologically, it has been proven that the quasi-2-normed spaces are Hausdorff topological vector spaces upon the usual topology (a topology generated by a family of balls). Another topology by a separating family of a-quasi-norms has been defined, and its coincidence with the usual topological structure has been demonstrated. Throughout this study, the sufficient and necessary assumption for the normality of quasi- -normed spaces, and the sufficient assumption for the metrizability property, have been proven