Research On Monad Of A Filter Theory And Applications To Uniform Space

In this paper, the concept of a filter is given by neighborhood system and the clas-sification of filters is showed. The concepts of filters and ultrafilters are characterized by nonstandard analysis. The properties of filters and ultrafilters and their relationships are discussed. Further, the monad of a filter is defined and its properties are discussed. Finally, several properties of the monad of a uniformity are discussed and the completeness of uniform spaces is characterized by the monad of Cauchy filters.The main points of this paper are as follows:In the first chapter, the idea of nonstandard analysis is introduced. Then the back-ground, the development and the research states of nonstandard analysis are presented.In the second chapter, the concept and the classification of filter are showed at first. Some properties of filters and ultrafilters are given. Ultimately, a special kind of connection topologies are generated through filters.In the third chapter, the definition and properties of superstructure are discussed. The formal languages in mathematical logic and the transfer principle are presented. Finally, the properties of two kinds of classic nonstandard models are showed.In the fourth chapter, the monad of a filter is defined and the supremum and the infimum of a family of filters are discussed. The concepts of ultrafilters are characterized by the monad of a filter. Based on these concepts, the nonstandard characterizations of filters are showed according to the general relationships between filters and ultrafilters. Finally, some properties of nonstandard compactification of topological spaces and its quotient spaces are studied.In the fifth chapter, uniform spaces are studied by monads of filters. The nonstandard characterizations of uniformity and properties of monads of uniformity are introduced first. Then, monads of Cauchy filters in uniform spaces are studied and the completeness of uniform spaces is characterized by monad of Cauchy filters.