Originally Posted by

**Aki** Yes.

But, I'm not sure about the converse too.

What troubles me is the following.

Given a compact set $\displaystyle K$ in $\displaystyle X$ and an open covering $\displaystyle \{U_\lambda\}_{\lambda\in\Lambda}$ of $\displaystyle q^{-1}(K)$,

since $\displaystyle \{q(U_\lambda\)\}_{\lambda\in\Lambda}$ is an open covering of $\displaystyle K$, one can choose from it a finite covering $\displaystyle \{q(U_\lambda_i)\}_{i=1,\cdots,N}$ of $\displaystyle K$. But, $\displaystyle \{U_\lambda_i\}_{i=1,\cdots,N}$ is not necessarily a covering of $\displaystyle q^{-1}(K)$.