This is related to Forster Riemann surfaces Ex 5.7 of Chpt 1.
Suppose $X,Y$ are connected hausdorff spaces. Show that every 2 sheeted covering map $p:Y\to X$ is galois.
Q: Why would be a 2 sheeted covering necessarily be connected here? I can totally have $Y=X\sqcup X$ which is disconnected as $X$ homeo to $X$ itself and thus $Y$ is basically made of two disconnected components. I think my intuitive picture is wrong here.
The reason asking this question is not to look for solution but why I hope $Y$ connected here in general.(Pictorially, I would hope $Y$ looks like $X\sqcup X$. Correct my picture if I am wrong here.) I knew $Y$ being connected used in the proof to show $g\in Deck(Y/X)$ $\{y\in Y\vert g(y)=y\}\sqcup\{y\in Y\vert g(y)\neq y\}$ open disjoint set to see contradiction.
The connectedness of $Y$ is part of the hypothesis.Otherwise, for a covering map to be Galois, the covering space MUST be connected, by definition.