线性序拓扑空间中的子空间与子序空间

Subspace of Linear Ordered Topological Space and Sub-ordered Space

R.Engelking在《General Topology》中讨论了线性序集的序拓扑的子空间和子序空间的关系,指出两种子空间是不同的,并给出了它们同胚的一些充分条件。本文给出了它们同胚的充要条件;证明了任何可数线性序空间与有理数的某个子空间同胚,且举例说明对非可数线性序集并没有类似结果。最后证明了良序集和实数集合具有序拓扑遗传性。

In'General Topology', R. Engelking has discussed the relation between ordered topological subspace of linear ordered set and sub-ordered space.He has pointed out that they are different and given some sufficient condi-tions for their homeomorphism.In this paper we first give the sufficient and necessary condition for their homeomorphism and prove that every countable linear ordered topological space is homeomorphic to some subspace of rational numbers and exemplify that the result for incountable linear ordered set is not the same. Finally we prove that well-ordered set and real set both have ordered topological hereditary.