New non-standard topologies

In this paper, on a non-standard extension of a metric space , we construct a chain of new non-standard topologies in terms of convex subrings of , its minimal element is the -topology and its maximal is the -topology. Next, we construct , the -asymptotic hull of , and we prove that such space is metrizable and complete when is generated by an asymptotic scale. Finally, we provide a pseudo-valuation taking integral values, equivalent to the classical Robinson's valuation, on , the Robinson's field of -asymptotic numbers.