  The paper introduces the class of O-metric spaces, a novel generalization of
metric-type spaces, classifying almost all possible metric types into upward
and downward O-metrics. We list some topologies arising from O-metrics and
discuss convergence, sequential continuity, first countability and T$_2$
separation. The topology of an O-metric space can be generated by an upward
O-metric on the space hence the focus on upward O-metric spaces. A theorem on
the existence and uniqueness of a fixed point of some contractive-like map is
proved and related with some other well known fixed point results in
literature. Applications to the estimation of distances, polygon inequalities,
and optimization of entries in infinite symmetric matrices are also given.
