I'm studying Hatchers 'algebraic topology', but I'm stuck on the "orientability of a manifold" part.
The author introduces the orientable double cover $\tilde{M}$ of a manifold $M$ in chapter 3, right above Proposition 3.25.
What I cannot understand is the orientability of the double cover $\tilde{M}$. I understood how to construct the local orientations $\tilde{\mu_{x}}$ of $\tilde{M}$ at $\mu_{x} \in U(\mu_{B})$, and how they come from the local orientation $\mu_{B} \in H_n(M\mid B)$.
But how can I show that the local orientations $\{ \tilde\mu_{x}\}_{x \in M}$ satisfy the "local consistency condition"?
(i.e. both $\tilde\mu_{x}, \tilde\mu_{x} \in U(\mu_{B}) \text{ come from the same generator in } H_n(\tilde M \mid U(\mu_B)$)
I would be grateful for any help, thanks :)