ABSTRACT

Beside using nonstandard analysis as independently developed field of

mathematics in general and especially in mathematical Analysis, the using of

nonstandard as techniques have more powerful for introducing and developing

different concepts in mathematics.

In this thesis, we use some concepts of nonstandard analysis given by

A. Robinson, (Robinson, 1974) and axiomatized by E. Nelson, (Nelson, 1977)

to reprove some classical results of Compactness by using the nonstandard

tools and get some new results. Also, we introduce a nonstandard definition of

quasi-metric spaces and quasi-compactness and generalizing these concepts to

be valid for external sets.