浅谈实数域上的拓扑空间

在实数域上,构造了三个拓扑空间:左闭拓扑空间E、右闭拓扑空间G、无限拓扑空间F。左闭拓扑空间E的基为{Br(z,d)|r>0为有理数,z∈R},满足第二可数公理。在分离性上,左闭拓扑空间E既是正则空间,又是正规空间,从而是T2空间、T1空间、T0空间,但是是非连通的。右闭拓扑空间G结构和左闭拓扑空间E相同。无限拓扑空间F是T0空间,不是T1空间。但是,无限拓扑空间F是连通的。

Simple Discussion on the Topological Space on the Real Number Field

In the real line,three topological spaces,which are the left shut topological space,the right shut topological space and the limitless topological space,are constructed.For{ B_r(z,d)| r>0,r∈Q,z∈R} is a basis to the left shut topological space,it satisfies the second countable axiom.In the separating aspect,it is a Normal space and Regular space,which is also a Hausdorff space and T_1 space,but it is not connected.The right shut topology has the same topological structure.The limitless topology is a T_0 space,but it is not a T_1 space.However,it is connected.