关于σ-空间的基和极小基

On bases and minimal base forσ-Space

基是拓扑空间中的研究热点.首先,在σ-空间的基础上给出了σ-空间基的定义.接着,借助例题阐述σ-基和拓扑基之间的联系,进一步得到P(X)的任何子集都是X上的某些σ-结构的基.其次从基的角度讨论σ-连续、(σ1,σ2)-连续等性质.最后分析σ-空间中极小基的存在性问题,结论说明极小基存在当且仅当σ-基中不存在并可约元,以及有限σ-空间的极小基是唯一的.

Base is a hot topic in topological space.First of all,the definition ofσ-base is given,then the relationship between the notion ofσ-base and topological base is illustrated with an example.Furthermore,any subset of P(X)is the base of someσ-structure on X.Secondly,the properties ofσ-continuity and(σ1,σ2)-continuity are discussed from the perspective of base.Finally,the existence of a minimal base inσ-space is analyzed.The results show that a minimal base exists if and only if there is no reducible element in theσ-base.And the minimal base is unique for a finiteσ-space.