= Solution
For every oriented open set $U$ and nowhere-vanishing <orientation> form $\omega$, let $\widehat U_\omega$ contain the points $(p,o_p)$ for which $o_p$ is the <orientation> determined by $\omega_p$. These sets form a <basis of a topology> for the desired topology on the <orientation double cover>.
They cover the set of all pairs: every point has an oriented <coordinate chart>, and changing the sign of its <orientation> form gives the other <orientation>. To check the basis intersection axiom, consider $\widehat U_\omega$ and $\widehat V_\eta$. On $U\cap V$ there is a unique smooth nonzero function $h$ with $\eta=h\omega$. Their intersection is the sheet over
$$
W=\{p\in U\cap V:h(p)>0\}.
$$
Since $W$ is open, this intersection is $\widehat W_{\omega|_W}$, another basis set. Thus intersections are open without requiring the two local orientations to agree on every component of the overlap.
The restriction $\pi:\widehat U_\omega\to U$ is bijective. Its inverse sends a relatively open set $V\subset U$ to the basis set $\widehat V_{\omega|_V}$, so the inverse is continuous. Conversely the projection of the intersection of $\widehat U_\omega$ with any other basis set is an open agreement set like $W$, so the projection is continuous as well. Hence \b[each sheet projects homeomorphically onto its oriented open set]. In particular,
$$
\boxed{\pi^{-1}(U)=\widehat U_\omega\sqcup\widehat U_{-\omega},}
$$
with both sheets open. This establishes the two-sheeted <covering space> structure.
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