函数空间上的Cauchy收敛拓扑

摘要

This article investigates the equivalent conditions of Cauchy convergence topology Tc.u,point-wise convergence topology Tp,compact-open topology Tk and uniform convergence topology Tu of continuous function space C(X,R) from metric space X to real line R.By the Cauchy cover,it obtains a dual theorem on the character of (C(X,R),Tc.u) and Cauchy cover number of X,proves that (C(X,R),Tc.u) is metrizable if and only if (C(X,R),Tc.u) is first countable if and only if Cauchy cover number of X is countable,affirmatively answers a question of Michael H Clapp's.%本文研究了度量空间X到实直线R上的连续函数空间C(X,R)上的Cauchy收敛拓扑Tc.u,点态收敛拓扑Tp.u,紧开拓扑Tk和一致收敛拓扑Tu相等的等价条件.利用Cauchy覆盖得到了(C(X,R),Tc.u)的特征与X的Cauchy覆盖数相等的一个对偶定理,获得了(C(X,R),Tc.u)可度量化当且仅当(C(X,R),Tc.u)是第一可数的当且仅当X具有可数Cauchy覆盖数,肯定地回答了Michael H Clapp等在文献[1]中提到的问题.