Context: on p235 of Hatcher's Algebraic Topology, Hatcher proves that if an n-manifold $M$ is connected, it is orientable if and only if its orientable double-cover $\tilde{M}$ has two components.

The first two lines of the proof are only a statements about covering spaces, but they puzzle me. If we have an 2-sheeted covering space $\tilde{X} \to X$, and $X$ is a connected space, why is it true that $\tilde{X}$ can only have one or two components? I tried to do a proof by contradiction to convince myself, but cannot come up with a suitable one.

The second problem I'm facing is with the idea conveyed by the second sentence of the proof, which seems to claim that each of the components is mapped homeomorphically to all of $X$.

Where are these claims coming from? Are they obvious?


Observation: By path lifting property, you will find that any two points connected by a path on the manifold $M$ will be lifted to a path, with a chosen starting point at $\tilde{M}$. So now the whole lifted path stays in exactly one component of $\tilde{M}$.

This means, once you have a preimage $\tilde{p}$ of a point $p \in M$ in a component of $\tilde{M}$, then by connectedness of $M$, since $\forall q$ can be connected by a path from $p$, we will see in this component we can find a complete preimage of $M$ there.

Now you can use the hypothesis that $M$ is two-sheeted. details are left for you, Acton.


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