Non-standard Analysis Method In Fuzzy Topology In Certain Application

Nonstandard analysis is the new branch to research various mathematical prob-lems using nonstandard models. Since nonstandard analysis is founded by A. Robin-son in1961, the field of real numbers and various relations over it are called the standard model. In the standard model, or in the reals, analysis is said to be stan-dard analysis. And the enlargement of the reals and the relations over it are called the nonstandard model. In the nonstandard model, or in the hyperreals, which is the extension of the reals, analysis is said to be nonstandard analysis. Nonstandard analysis is the true extension of standard analysis. The difference between them is the infinitesimal and the infinite.The research mainly studies the nonstandard model theory and some applica-tions in fuzzy topology. There are two purposes in this research. One is to convert the nonstandard model into a superstructure, which can convenient for the twice model, even the multi-model. The other is to provide the method for the combi-nation of nonstandard analysis and fuzzy topology. These attempts not only can improve nonstandard analysis, enriching its applications, but also can provide a new method to study fuzzy topology. The main conclusions in this research are as follows:(1) The individuals set S of a class and a superstore V(S) on it are given, and it is shown that V(S) is large enough. Founded first-order language in the math-ematical logic, the concepts of abstract nonstandard model, specific nonstandard model and pre-nonstandard model are defined with the interpretation mapping in model theory. A new superstructure V(*S) is constructed with ultrapower, and it is proved that V(*S) is a nonstandard model of V(S) by ultrafilter. The necessi-ties of ultrafilter and bounded quantifiers sentence are shown in transfer principle. The structure and properties of mapping*in nonstandard analysis are discussed from several views. Thus, from a superstructure to a new superstructure, it is the foundation of the twice model or the multi-model.(2) The standard entities, internal entities and external entities in V(*S) are studied. It is shown that*V(S) is only a symbol, which is not the*-image of V(S). The existence ofκ-enlarged model is proved by κ-adequate filter, and the sufficient and necessary conditions and some interesting properties forκ-enlarged model are obtained. Similar to Luxemburg's way, the existence, necessary conditions and some properties ofκ-saturated model are discussed. Based on these, the existence of the twice model V(**S) is proved. Furthermore, the enlargement and saturation for the twice model are discussed with the entities in it, such as the twice standard entities, the twice internal entities, etc. The basic ideas and methods are provided, in this part, for the multi-model in future.(3) The relations between a complete lattice L and its nonstead extension*L are discussed. The concepts of internal complete lattice, the κ-copy complete lattice and κ-complete lattice are defined. Especially, K-complete lattice is very important for fuzzy topology. For any nonempty subset A, which cardinal less than κ, in a lattice L, if both∨A and∨A exist, L is called κ-complete lattice. Based on it, the representations and properties of the enlarged model and the saturated model are obtained in fuzzy forms. It is a basic way to further research.(4) The nonstandard fuzzy sets are defined, and some properties of the family of all nonstandard fuzzy sets*[0,1]*x and its subfamilies are discussed, such as standard fuzzy sets°[0,1]x, internal fuzzy sets*[0,1]*x, etc. With standard part mapping st, the relations between fuzzy sets and nonstandard fuzzy sets are shown. Then the nonstandard ultra-compactification (*X, δ) of a fuzzy topological space (X,δ), even Stone-Cech ultra-compactification (*X,δ) are obtained by a nature way.(5) Based on the interval numbers Ⅱ(R) and its order, the hyperinterval numbers Ⅱ(*R) are provided. And an equivalence relation～is defned in Ns(Ⅱ(*R)), which is subset of Ⅱ(*R), and it is proved that Ⅱ(R) and N(Ⅱ(*R))/～are orderi-somorphic. Then a nonstandard extension (*X,*p) of an interval numbers valued metric space (X, p) is discussed. It is an hyperinterval numbers valued metric space. Furthermore, the completion of (X, p) is obtained by nonstandard points in (*X,*p), finite points Fin(*X) and qusi-near standard points Qns(*X).(6) The monad m(?) of a fuzzy filter on X X X is defined. And a sufficient and necessary condition is shown when a fuzzy filter on X x X is a fuzzy uniformity on X. It is that m(?) is a fuzzy equivalence relation on*X. The nonstandard ex-tension (*X,(?) of fuzzy uniform space (X,?) and its properties are discussed. It is proved that (*X,?) is a nonstandard fuzzy uniform space, and it is nonstandard ultra-completion of (X,(?)...