| A pre-topological group is a group endowed with pre-topology such that the inverse mapping and product mapping are pre-continuous.This paper mainly studies some properties of pre-topological groups and some generalizations of pre-topological groups.In Chapter 1,we introduce the sources of pre-topological groups and the research status at home and abroad.At the same time,we introduce some related definitions and theorems of pre-topological groups.In Chapter 2,we define three pre-topological groups with different properties,that is,symmetrically pre-topological group,strongly pre-topological group and almost topological group.In addition,we give some generalizations of pre-topological group,such as semi-pre-topological group,quasi-pre-topological group,para-pre-topological group etc.By enumerating some examples to illustrate their existence and proving their relationship.In Chapter 3,we mainly study the topological properties of pre-topological groups and co-reflexion topological groups,such as separability,connectedness,compactness,axioms of countability and some other topological properties.We prove that the co-reflexion topological group is regular if and only if the pre-topological group is regular.In Chapter 4,we study some topological properties of pre-topological groups.It is mainly proved that each T0 pre-topological group is regular and every almost topological group is completely regular. |
