Contraction of a Generalized Metric Structure

Details Contraction of a Generalized Metric Structure Contraction of a Generalized Metric Structure

2008-12-10

arxiv.org/abs/0812.1898v1

Mathematics

Geometric Topology

20 pages

Scientific paper

In some scientific fields, a scaling is able to modify the topology of an observed object. Our goal in the present work is to introduce a new formalism adapted to the mathematical representation of this kind of phenomenon. To this end, we introduce a new metric structure - the galactic spaces - which depends on an ordered field extension of R. Moreover, some natural transformations of the category of galactic spaces, the contractions, are of particular interest: they generalize usual homotheties, they have a ratio which may be an infinitesimal, they are able to modify the topology and they satisfy a nice composition rule. With the help of nonstandard extensions we can associate to any metric space an infinite family of galactic spaces; lastly, we study some limit properties of this family.

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