Ideal Mappings between Ideal Spaces: Contrast, Exhaustiveness, Correspondence, and Homeomorphism

Abstract

 Ideals of subsets can be used in a theory of ideal topological spaces, which generalizes classical topological concepts. Several notions of continuity and related mappings have been studied in ideal spaces, but there is the need for a unified definition of mappings by ideal sets, which is not using the notion of continuity. In this paper, several kinds of mappings between ideal spaces are introduced and investigated: ideal contrast mappings, ideal exhaustive mappings, and ideal correspondence mappings. Further, the notion of mappings which are ideal continuous is also taken into consideration, and ideal homeomorphisms are described with the help of the introduced mapping properties. The basic goal is to build basic relationships between these mappings and to determine sufficient conditions for the preservation of their properties. Various propositions and theorems are set up such as characterizing ideal homeomorphisms with bijective-ideal correspondence mappings. The results show that the concepts introduced here offer a systematic way of studying mappings between ideal spaces, while only the ideal structures related to the sets on which they operate are needed. The results are used to advance the theory of ideal topological spaces, and serve as a basis for further studies of generalized mappings and structural relationships between ideal spaces.