The Study On A Class Of General Paracompact Spaces

Compactness is very important in topology which is the main study of topological invariant properties.And many scholars have achieved remarkable fruitful results in the field of compactness theory.The theory of compactness,expansibility and closed space of compactness theory is very important.So the study on this is significant:First,this paper defines q-(countable)expandable space,and some properties are obtained under the q-condition of keeping the closure.Then,the relation between q-expandable space and ?-q-countable expandable space is given.Under the condition of extremely disconnected,the relation between the q-expandable space and some other expansion spaces is given.Second,this paper defines q-(countable)imitated compact space,yq-imitated compact subsets and ?q-closed set,and obtain their correlation with the compactness space.On this basis,the q-I compactness space and q-?imitated compact space are defined in the topological space,and they have the property of closed heredity.Then,the paper gives the relations between Q regular space,strong Q regular space,strong Q normal space and q-?compactly compact spaces.And then,the paper defines the full subcategory and product and coproduct,and further get the category of q-? compactly compact space and q-? compactly compact space.Third,Q-closed space and weak Q-closed space are defined in topology;It is concluded that Q-closed space is equivalent to weak Q-closed space.The relation between S-closed space,Q-closed space and weak Q-closed space is obtained.Fourth,this paper defines q-paracompact space with q open set and star refinement sequence,and further proves that they are equivalent.