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For a coordinate chart $x:U\to x(U)$ and either orientation sheet over $U$, take the chart $x\circ \pi$ on that sheet. On a nonempty overlap, its transition to $y\circ \pi$ is exactly $y\circ x^{-1}$, restricted to the open agreement set from part (a). It is smooth with smooth inverse. These charts therefore define a smooth structure for which $\pi$ is locally a diffeomorphism.

The topology is Hausdorff: points with distinct base points are separated by inverse images of disjoint base neighborhoods; two points over the same base point are separated by the two disjoint sheets over one coordinate neighborhood. A countable coordinate cover of $M$ gives a countable collection of two-sheeted charts, whose Euclidean chart bases show second countability. Thus the cover is a genuine smooth manifold, not just a collection of compatible charts. The construction using all base charts makes the smooth structure independent of the chosen atlas. It also makes $\pi :{\hat{U}}_{\omega }\to U$ a diffeomorphism for every oriented open $U$, since it is a bijective local diffeomorphism.

At $\hat{p}=(p,o_p)$, the derivative $d{\pi }_{\hat{p}}:T_{\hat{p}}\hat{M}\to T_{p}M$ is a linear isomorphism. Define the orientation of $T_{\hat{p}}\hat{M}$ to be the one mapped to $o_p$. On ${\hat{U}}_{\omega }$ it is represented by ${\pi }^{\ast }\omega$. Where two sheets overlap, their base orientation forms differ by a positive function, so their pullbacks determine the same orientation. This proves the canonical orientation of the cover:The deck involution $(p,o_p)\mapsto (p,-o_p)$ reverses this canonical orientation of the orientation double cover. No global orientation of the base was used.