关于闭映射的两个定理

TWO THEOREMS ON CLOSED MAPPINGS

本文引入比正规和可数仿紧都弱的一个条件(*),证明以下两个定理:(1)设f:Z→Y是闭映射,Z是满足条件(*)的等紧空间,Y是T_2空间,则f是紧覆盖映射;(2)设f:Z→Y是闭映射,Z是满足条件(*)的等紧空间,Y是T_2Frechet空间,则存在闭子集Z′Z,使f|Z′:Z′→Y是既约映射.

Consider condition(*):for each countable discrete closed set {x_n:n∈N}, there exists a locally finite open collection {W_n: n∈N}such that for each n∈N. x_n∈W_n,. We prove the following two theorems: (l)If f : Z→Y is a closd mapping,Z is an isocompact space satisfying condition(*),Y is a T_2 Space, then f is a compact-covering mapping; (2) if f:Z →Y is a closed mapping,Z is an isocompact space satisfying condlition(*),Y is a T_2 Frechet Space,then there is a closed set Z'■Z such that f|_z: Z'→Y is irreducible.