Let $X$ be a space with a closure-preserving cover 
$\FF$ consisting of countably compact closed subsets.
In this paper we prove the following:
(1) if $X$ is normal and each $F \in \FF$ is weakly 
infinite-dimensional, then $X$ is weakly infinite-dimensional, 
(2) if $X$ is collectionwise normal and each $F \in \FF$ is 
a $C$-space, then$X$ is a $C$-space.