A collection $\cal P$ of subsets of $X$ is {\it point-countable} if every point of $X$ is in at most countably many elements of $\cal P$. Every first countable space having a countable open (or closed) cover of metric subsets need not be metrizable. For a space $X$ having a (not necessarily open or closed) point-countable cover of metric subsets, we shall consider conditions for $X$ to be metrizable in terms of weak topology.