序列覆盖的闭映射保持可度量性

Metrizability is Preserved by Sequence-Covering and Closed Maps

设f:X→Y是连续的满映射. f称为序列覆盖映射,若{y})是Y中的收敛序列,则存在X中的收敛序列{xn},使得每一xn∈f-1(yn);f称为1序列覆盖映射,若对于每-y∈Y,存在x∈f-1(y),使得如果{yn}是Y中收敛于点y的序列,则有X中收敛于点x的序列{xn},使得每一xn∈f-1(yn).本文研究度量空间序列覆盖的闭映射之构造,否定地回答了Topology and its Applications上提出的一个问题.

Let f : X →Y be a continuous and surjective map. f is a sequence-covering map if whenever {yn} is a convergent sequence in Y there is a convergent sequence {xn} in X with each xn ∈ f-1(yn). f is a 1-sequence-covering map if for each y ∈ Y, there is x ∈ f-1(y) such that whenever {yn} is a sequence converging to y in y there is a sequence {xn} converging to x in X with each xn ∈ f-1(yn}. In this paper the structure of sequence-covering and closed maps of metric spaces is investigated, a problem posed by 'Topology and its Applications' is negatively answered.