How to show explicitely that 2-sheeted covers are Galois?

Question

Let $X,Y$ be connected Hausdorff topological spaces. It is well-known that every 2-sheeted covering $p:Y\to X$ is Galois which means that $Aut(Y/X)$ acts transitively on fibers. It is easy to come up with a $g\in Aut(Y/X)$ acting transitively on fibers. Namely, for $y\in Y, p^{-1}(p(y))=\{y,c\}$ and we let $g(y)=c$. The problem is that I don't know how to show explicitely that $g$ is continuous. Namely, showing explicitely that for every open set $V$, $g^{-1}(V)$ is open.