Abstract
In this paper, on a non-standard extension (*X, *d) of a metric space (X,d), we construct a chain of new non-standard topologies in terms of convex subrings of *ℝ, its minimal element is the S-topology and its maximal is the Q-topology. Next, we construct X, the F-asymptotic hull of X, and we prove that such space is metrizable and complete when F is generated by an asymptotic scale. Finally, we provide a pseudo-valuation taking integral values, equivalent to the classical Robinson's valuation, on ρ ℝ, the Robinson's field of ρ-asymptotic numbers.
| Original language | English |
|---|---|
| Pages (from-to) | 323-344 |
| Number of pages | 22 |
| Journal | Monatshefte fur Mathematik |
| Volume | 172 |
| Issue number | 3-4 |
| DOIs | |
| State | Published - Dec 2013 |
Bibliographical note
Funding Information:The research has been funded by the King Fahd University of Petroleum and Minerals project IN111038.
Funding Information:
I thank the referee for insightful comments which led to significant improvements in the paper. Also, I would like to acknowledge the support provided by the Deanship of Scientific Research (DSR) at the King Fahd University of Petroleum and Minerals (KFUPM) for funding this work through project IN111038.
Keywords
- Non-standard analysis
- Q-topology
- Robinson's asymptotic numbers
- S-topology
ASJC Scopus subject areas
- General Mathematics