Nonstandard Completion of Non-complete Metric Spaces

Nonstandard Completion of Non-complete Metric Spaces

By: Ala Omer Hassan

ABSTRACT

Standardly, to prove that a metric space is complete, it is equivalence to proving that every Cauchy sequence in  is convergent to a point in it. There are a lot of spaces or sets  which agree with this property except at a small size subset of . In this thesis, by using nonstandard analysis tools found by (A. Robinson, 1974) and axiomatized by (E. Nelson, 1977), we try to reformulate the definition of completion corresponding to nonstandard modified metric , and to give a nonstandard form to the classical (standard) completion theorem and to use the power of nonstandard tools to overcome the incompetence of those spaces which have deprivation at a small size subset. In addition, we construct new nonstandard definitions nearly complete, nearly dense and nearly isometry and using them to prove results that we get in this thesis. Now, we indicate to the main of these results that we obtain:

  • Let be a metric space and let be the set of all Cauchy sequences in . The two sequences  and inare said to be:
  • In the same order and denoted by , if
  • Standardly in the same order denoted by , if
  • Unlimitedly in the same order denoted by , if

 

  • Letandbe two metric spaces such that is standard. Then a function  is nearly isometry if and only if for all  there exists  such that .
  • is complete metric space, or there is a nonstandard completion  for a non-complete metric space (.
  • If is nearly complete, then  is completion of , and either  or .