Categorical Properties Of θ-open Set,δ-open Set And Three Kinds Of Special Closure Spaces

θ-open set andδ-open set are special open sets. In this paper, def-initions ofθ-space,δ-space, weaklyθ-continuous mapping,θ-continuous mapping, stronglyθ-continuous mapping, weaklyδ-continuous mapping,δ-continuous map-ping, stronglyδ-continuous mapping are presented, and their corresponding proper-ties are discussed. In addition, some categorical corresponding properties relationed withθ-open set andδ-open set are studied. Otherwise, the notions of closure space, bottomless closure space, algebraic closure space and reasoning closure space are introduced and their categories corresponding properties are studied.The key points and the main contents of this paper are as follows:In chapter 1, some concepts and important conclusions which are related toθ-open set,δ-open set and three kinds of special closure spaces are introduced.In chapter 2,using the way of general topology, weaklyθ-continuous mapping,θ-continuous mapping, stronglyθ-continuous mapping, weaklyδ-contin-uous mapping,δ-continuous mapping, stronglyδ-continuous mapping are defined first. Then the relationships among them are studied. Finally, properties of these mapping are given.In chapter 3. firstly, the categories Topsθ(the category of all topological spaces and stronglyθ-continuous mappings), Top (the category of all topological spaces and continuous mappings). Topθ(the category of all topological spaces andθ-continuous mappings), Topsδ(the category of all topological spaces and stronglyδ-continuous mappings), Topδ(the category of all topological spaces andδ-continuous mappings),θTop (the category of allθ-topological spaces and continuous mappings),δTop (the category of allδ-topological spaces and continuous mappings) and Reg (the category of all regular spaces and continuous mappings) are introduced,then their properties and relationships among them are studied. Secondly, the categories CS (the category of all closure spaces and continuous mappings). NCS (the category of all bottomless closure spaces and continuous mappings), ACS (the category of all algebraic closure spaces and continuous mappings) and RCS (the category of all reasoning closure spaces and continuous mappings) are introduced,then the prop-erties of NCS,ACS and RCS are shows, and the relationships among them are studied.