摘要
利用狭义拟仿紧空间的等价刻划，给出了一个非狭义拟仿紧的正规弱θ－加细空间，此外还证明了强完备映射的逆保持狭义拟仿紧性．这两个结果分别回答和部份回答了蒋继光提出的两个问题．
Strong quasiparacompactness was first introduced by Liu in . It is known that θrefinabilitystrong quasiparacompactness weak θrefinability, and the first implication is not reversible. In , Jiang raised following two questions.Question 1. Does there exist a weak θrefinable, Hausdorff or regular space which is not strongly quasiparacompact?Question 2. Do perfect mappings inversely preserve strong quasiparacompactness?In this paper, we obtain some characterizations of strong quasiparacompactness, and answer Question 1. For Question 2, We give a partial answer. The main results are as follows.Definition 1. A space X is called strongly quasiparacompact if for every open cover U of X there exists a refinement {βn:n<ω0} of U such that βn is a relatively discrete closed collection in X∪i<nβ*i. If also ∪i<nβ*i is closed in X for every n<ω0 then we call X is B(D,ω0)refinable.Definition 2. A space X is called weak θrefinable (boundly weak θrefinable) if for every open cover U of X there exists an open refinement ∪n<ω0Un of U such that the following conditions (1) and (2) ((1) and (2)) hold:(1) for each x∈X, there exists n<ω0, such that 0<ord(x,Un)<ω0(1) there exists k<ω0, such that 0<ord(x,Un)≤k for each x∈X and some n<ω0, where k is called bound.(2) the open cover {U*n:n<ω0} is point finite.Definition 3. A subset F of a space X is called boundly mcompact if for every open collection U of X, where U covers F, there exists a subcollection U of U, such that U covers F and |U|≤m. A closed mapping f:X→Y is called a strongly perfect mapping, if there exists m<ω0 such that for each y∈Y,f-1(y) is a boundly mcompact subset of X.Example. There exists a normal weak θrefinable space, which is not strongly quasiparacompact.Theorem 1. A space X is strongly quasiparacompact if and only if X is B(D,ω0)refinable.Theorem 2. A space X is strongly quasiparacompact if and only if X is boundly weak θrefinable, where the bound is 1.Theorem 3. If f:X→Y is a strong perfect mapping and Y is strongly quasiparacompact, then X is strongly quasiparacompact.
出处
《南京大学学报（自然科学版）》
CAS
CSCD
1998年第1期16-20,共5页
Journal of Nanjing University（Natural Science）
关键词
弱θ-加细
强完备映射
狭义拟仿紧空间
Strongly quasiparacompact
weak θrefinable
strongly perfect mapping.
