Covering space of disconnected spaces

Question

Let $f:M\to N$ is a $d-$fold covering. Is it true that $M$ is disconnected if $N$ is disconnected? If not, can anyone give a counter example?

Answer 1

Yes, if $N$ is disconnected so is $M$. This is a general case of the result "Let $f :M \to N$ be a continuous surjection, and $N$ disconnected. Then $M$ is disconnected."

To prove this, decompose $N$ as $N=U\cup V$ for non-empty disjoint open sets $U,V$.

Then $M=f^{-1}(U) \cup f^{-1}(V)$ and $f^{-1}(U), f^{-1}(V)$ are both

1. non-empty (since $f$ is a surjection)
2. disjoint (because $U$ and $V$ are)
3. open (because $U$ and $V$ are, and $f$ is continuous).