The Point Set Property And Continuity Exploration Of Two Kinds Of Semi-topological Spaces

The Hungarian mathematician A.Csaszar introduced the concept of generalized topology and gained a lot of research results in 2002.The generalized topology is actually a semi-topology,usually called the upper semi-topology.Therefore,scholars automatically introduce the lower semi-topology,left semi-topology?L-semi-topology?and right semi-topology?R-semi-topology?,and conduct some studies for these semi-topology.As a result,a series of results are obtained.In this thesis,we first introduce the properties of point set,definitions and properties of continuous mapping in generalized topological space.Then,after making an analogy between the methods of general topology and generalized topology,we study the R-semi-topology and L-semi-topology.The work of the following three aspects is mainly made.First,by reference to the techniques of general topology,we get several results in R-semi-topological spaces as follows:?1?If A is an open set,thenA?28?^oA.?2?If A?Y?Xand Y is the open subset of X,then A is open to Y if A is open to X.?3?Let ₁?,?₂ be two R-semi-topology in X,if?₁??₂,then?1?x?X,we have?₁?7?x?8???₂?7?x?8?;conversely,the conclusion is not established.?4?Let?7?X,??8?be a topological space,andbe a base of?,thensatisfies:?=X,but?=X is not necessarily established in the R-semi-topological space.Finally,we discuss the separation properties of R-semi-topological space,and put forward some results.Second,after comparing the continuity of mapping in general topology,we put out the definitions of reverse opening continuity and pointwise continuity in R-semi-topological space,and find that reverse opening continuity must be pointwise continuity,but pointwise continuity is not necessarily reverse opening continuity.Furthermore,through the introduction of R-neighborhood system,the definitions of continuity,weak continuity,almost continuity and strong continuity in R-neighborhood system are given.By discussing the relationships among them,we get the following results:?1?In R-neighborhood system,strong continuity is strictly stronger than continuity,continuity is strictly stronger than weak continuity.?2?Strong continuity is strictly stronger than almost continuity,almost continuity is strictly stronger than weak continuity.?3?Almost continuity is independent of continuity.Third,the difference between the properties of point set in two classes of semi-topological space is studied.It proves that:?1?The necessary and sufficient condition that A be an open set thatA?28?^oAbe unfounded in semi-topological space but it is established in L-semi-topological space.?2?In L-semi-topological space,A?Y?X,and Y is the open subset of X.If A is open to X,then A is open to Y.?3?In R-semi-topological space,A?Y?X,and Y is the closed subset of X.If A is closed to Y,then A is closed to X.Additionally,some results that two classes of semi-topological space are both founded are provided.