Monads in General Topology

Abstract

Within the set-theoretic stance on mathematics, at the beginning of the XX century a universal approach was developed to study the structure of continuity and proximity which was formulated in general topology. When considering the microstructure of the numerical line we have already seen that from the viewpoint of nonstandard analysis a set of infinitesimals arises as a monad, i.e., the external intersection of standard elements of the filter of zero neighbourhoods of the only separated topology that agrees with the algebraic structure of the field of real numbers. One can say that through the notion of the monad of a filter a certain synthesis of general topological and infinitesimal ideas is implemented, the corresponding relations being the basic objects of investigation of the present chapter. We will focus our attention on the most elaborated ways of studying classical topological concepts and constructions that group around compactness which is allowed into the nonstandard set theory through idealization. The contribution of the new approach to the problem under discussion is basically associated with the elaboration of a new principally important notion, that of a nearstandard point. The corresponding criterion of compactness of a standard space, i.e., the nearstandardness of its every point, demonstrates the value and essence of the concept of nearstandardness which carries out a certain individualization for the points of the conventional notion of compactness pertaining to sets. Similar techniques of individualization comprise an important and characteristic part of the arsenal of the nonstandard methods of analysis.