Dual Scott Topology On A Pre-set Sequence

In this thesis, we investigate the bi-Scott topology on preordered sets.In Chapter1, we introduce some necessary preliminaries and some results about the preordered sets.In Chapter2, we have three sections:In Section1:we define the bi-Scott topology on the preordered sets and discuss the properties of the bi-Scott open sets; we also define the new relation "(?)" and the bi-continuous preordered sets, and discuss some properties of the bi-Scott open and closed sets on bi-continuous preordered sets.In Section2:we define the bi-Scott continuous mappings, discuss the relationship between the bi-Scott continuous mappings and the Scott continuous mappings and we give some necessary and sufficient or necessary conditions of the bi-Scott continuous mappings.In Section3:we define quotient sets and quotient topology, and define partial order relation and bi-Scott topology on quotient sets; we discuss the relationship between the category POBSET of bi-Scott topological spaces on preordered sets and the category OBSET of bi-Scott topological spaces on posets, and prove that the category OBSET is the reflective subcategory of the category POBSET.