A generalized metric-type structure with some applications

The aim of this paper is to introduce a new class of metric-type spaces called O-metric spaces as a generalization of several metric-type spaces in literature, by constructing a triangle-type inequality that accommodates many binary operations including multiplication and division. Possible metric-type spaces are classified into upward and downward O-metric spaces as O-metrics between identical points are not necessarily minimal. Conditions for passage between upward and downward O-metrics are specified, giving rise to various reverse triangle inequalities. Topologies arising from O-metrics are listed, and properties such as O-convergence, sequential continuity, first countability and T2 separation are investigated. It is shown that the topology of an O-metric space can be generated by an upward O-metric on the space hence the focus will be on upward O-metric spaces. With the use of polygon inequalities, a theorem on the existence and uniqueness of fixed points of some contractive-like maps is established in the setting of O-metric spaces, and well known results are obtained as corollaries. Applications to the estimation of distances, polygon inequalities, and optimization of entries in some infinite symmetric matrices are also given.