Research On Some Properties In Semi-topological Spaces

In 2002,the Hungarian mathematician A.Csaszar introduced the concept of generalized topological space,and did research on the properties of point sets and subspaces in generalized topological space,and obtained countless results in this topological space.The generalized topology is actually a semi-topology.Therefore,in 2015,Hu Xichao et al.renamed the generalized topology as the upper half topology,and then introduced the concept of the lower half topology and obtained some basic results about the lower half of the topological space.Since then,many researchers have actively invested in re-dividing the topology into left-half topology and right-half topology,and obtained a series of results on these two types of semi-topologies.Based on the above research,this paper will further study the left half topology and the bottom half topology of the above four types of half topologies.The specific work and research results are as follows:(1)The related concepts in the analog topological space introduce the concepts of interior points,neighborhoods,cluster points,derived sets,topological bases in the left semi-topological(L-semi-topological)space,and discuss the properties of point sets in the L-semi-topological space,L-semi-topological comparison,L-semi-topological basis and continuity in L-semi-topological space,etc.,obtained some results about the properties of point sets in L-semi-topological space,and then carried out the thickness of L-semi-topology On the basis of comparison,several equivalent characterization theorems are obtained.Finally,the continuous mapping and separation properties in L-semi-topological space are studied,and relevant conclusions are obtained.This thesis through in-depth research on L-semi-topology,It provides a theoretical basis for future generations to continue learning,and also enriches the semi-topological theory.(2)By analogy with related concepts in topological space,we introduce related concepts such as interior points,neighborhoods,gathering points,derived sets,topological bases,and nets in the lower semi-topological((?)-semi-topological)space,and explore the(?)-semi-topological space The relationship between interior points,neighborhoods and derived sets;(?)-semi-topological comparison;J-semi-topological basis,J-semi-topological space and its convergence properties,etc.,and obtained a series of results.Finally,the T0,T1,T2,regularity and normality in the J-semi-topological space are also discussed,and the correspondingT0,T1,T2,regular and normal spaces in the J-semi-topological space are obtained.Several equivalent characterization theorems of,similarly,the article also demonstrates some propositions that are correct in general topological space but wrong in J-semi-topological space.