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.