I am looking for verification and/or criticism of a proof I have constructed. The problem is to show that every CW complex is compactly generated. A topological space $X$ is $compactly$ $generated$ if it has the property that for all subsets $A\subseteq X$ such that $A\cap K$ is closed in $K$, where $K\subseteq X$ is any compact subset, then $A$ is closed in $X$.
My proof: Let $A$ be a subset of $X$ such that $A\cap K$ is closed in $K$ for all compact subsets $K\subseteq X$. In particular $A\cap\bar{e}$ is closed in $\bar{e}$ for all cells $e$ in the cell decomposition. This is due to the fact that $\bar{e}$ is compact, being the image of $\overline{\mathbb{B}^n}$ under the characteristic map of $e$. Since $X$ is a CW complex it is coherent with the family $\{\bar{e}:e\in \mathcal{E}\}$ ($\mathcal{E}$ is the set of cells in the cell decomposition). Coherency along with the fact that $A\cap\bar{e}$ is closed in $\bar{e}$ for all $e$ directly implies that $A$ is closed in $X$.
Is this correct? Thanks.