On spaces having the weak topology with respect to closed coverings, II

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: