Let $f : M \mapsto N$ be continuous. Consider a compact $K \subseteq M$.

Assume as given: $\forall B \subseteq N$ open, $f^{-1}(B)$ is open in $M$. 

1. Consider an open cover of $f(K)$, call it $A\_i$ for $i \in I$.
2. Then, $B\_i$ is an open cover of $K$, where $B\_i = \{x \in M: f(x) \in A\_i \}$
3. There exists a finite subcover of $K$, call it $B\_{n\_1}, ..., B\_{n\_k}$
4. Then, $A\_{n\_1}, ..., A\_{n\_k}$ is a finite subcover of $f(K)$

Just need to verify this proof, in particular, I am concerned if $(2)$ and $(4)$ need more details.