On spaces having the weak topology with respect to closed coverings, II
On spaces having the weak topology with respect to closed coverings, II
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在具有相对于封闭覆盖物的弱拓扑的空间上，II
DOI：
10.3792/pja/1195525968
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发表时间：
1953
期刊：
影响因子：
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通讯作者：
Kiiti Morita
中科院分区：
文献类型：
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作者：
Kiiti Morita
在第一篇论文中[4]我们引入了以下概念。设 X 为拓扑空间，[A} 为 X 的闭覆盖。如果 [A} 的任何子集 [A} 的并集在 X 中是闭的，并且与每个 A 的交集相对于 A 的子空间拓扑开的任何 A 子集在子空间 A 中必然是开的，则称 X 具有相对于 [A} 的弱拓扑。任何 CW 复形（参见 [5]）相对于由所有单元的闭包组成的闭覆盖具有弱拓扑。作为另一个例子，我们注意到拓扑空间对于任何局部有限封闭覆盖总是具有弱拓扑。本文的目的是建立以下定理。定理1. 令X 为相对于闭覆盖[A}具有弱拓扑的拓扑空间。那么 X 是仿紧且正规的当且仅当每个子空间 A 都是准紧且正规的。因此，如果 X 具有相对于闭合覆盖 [A} 的弱拓扑，则所有子空间 A 的以下属性中的每一个都意味着 X 具有相同的属性：(1) 正态性，(2) 完全正态性，(3) 完全正态性，(4) 集合正态性，(5) 仿紧性和正态性，(6) 可数仿紧性和正态性。另一方面，所有 A 的局部紧性或可度量性并不一定意味着 X 具有相同的性质。 1. 引理 引理 1. 设 A 是准紧和正规空间 X 的闭子集。如果 {G} 是 A 中的局部有限系统，由 A 的开 Fo 集 G 组成，则存在 X 的开 Fo 集的局部有限系统 {H}，具有以下性质：
In the first paper under this title [4 we have introduced the following notion. Let X be a topological space and [A} a closed covering of X. Then X is said to have the weai topology with respect to [A}, if the union of any subcollection [A} of [A} is closed in X and any subset of A whose intersection with each A is open relative to the subspace opology of A is necessarily open in the subspace A. Any CW-complex (cf. [5) has the weak topology with respect o the closed covering which consists of the closures of all he cells. As another example we remark that a topological space has always he weak topology with respect to any locally finite closed covering. The purpose of this paper is to establish the following theorem. Theorem 1. Let X be a topological space having the weak topology with respect to a closed covering [A}. Then X is paracompact and normal if and only if each subspace A is pracompact and normal. Thus if X has the weak topology with respect to a closed covering [A}, each of he following properties for all subspaces A implies he same property for X: (1) normality, (2) complete normality, (3) perfect normality, (4) collectionwise normality, (5) paracompactness and normality, (6) countable paracompactness and normality. On the other hand, local compactness or metrizability for all A does not necessarily imply the same property for X. 1. Lemmas Lemma 1. Let A be a closed subset of a pracompact and normal space X. If {G} is a locally finite system in A which consists of open Fo-sets G of A, then there exists a locally finite system {H} of open Fo-sets of X with the following properties: