In this paper, on a non-standard extension \documentclass[12pt]{minimal}
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\begin{document}$$\displaystyle ( {}^*X, {}^*d)$$\end{document} of a metric space \documentclass[12pt]{minimal}
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\begin{document}$$\displaystyle (X,d)$$\end{document}, we construct a chain of new non-standard topologies in terms of convex subrings of \documentclass[12pt]{minimal}
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\begin{document}$$\displaystyle {}^*\mathbb{R }$$\end{document}, its minimal element is the \documentclass[12pt]{minimal}
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\begin{document}$$\displaystyle S$$\end{document}-topology and its maximal is the \documentclass[12pt]{minimal}
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\begin{document}$$\displaystyle Q$$\end{document}-topology. Next, we construct \documentclass[12pt]{minimal}
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\begin{document}$$\displaystyle \widehat{X}$$\end{document}, the \documentclass[12pt]{minimal}
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\begin{document}$$\displaystyle {\fancyscript{F}}$$\end{document}-asymptotic hull of \documentclass[12pt]{minimal}
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\begin{document}$$\displaystyle X$$\end{document}, and we prove that such space is metrizable and complete when \documentclass[12pt]{minimal}
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\begin{document}$$\displaystyle {\fancyscript{F}}$$\end{document} is generated by an asymptotic scale. Finally, we provide a pseudo-valuation taking integral values, equivalent to the classical Robinson’s valuation, on \documentclass[12pt]{minimal}
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\begin{document}$$\displaystyle {}^\rho \mathbb{R }$$\end{document}, the Robinson’s field of \documentclass[12pt]{minimal}
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\begin{document}$$\displaystyle \rho $$\end{document}-asymptotic numbers.
