How can I give an CW complex structure to the space $X$ using a covering map $p:X\rightarrow E$ where $E$ has structure of CW complex?

Question

I think that idea is to take an n-cell $X_{n}$ of space $E$ a proof $p^{-1}(X_{n})$ is a union of n-cells in $X$. I found in other post the idea is right, but I can't prove that, in my case n-cells are open, therefore I can't use compactness argument of cells. I really appreciate your help.