 Solution

For every oriented open set $U$ and nowhere-vanishing orientation form $\omega$, let ${\hat{U}}_{\omega }$ 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 ${\hat{U}}_{\omega }$ and ${\hat{V}}_{\eta }$. On $U\cap V$ there is a unique smooth nonzero function $h$ with $\eta =h\omega$. Their intersection is the sheet overSince $W$ is open, this intersection is ${\hat{W}}_{\omega {|}_W}$, another basis set. Thus intersections are open without requiring the two local orientations to agree on every component of the overlap.

The restriction $\pi :{\hat{U}}_{\omega }\to U$ is bijective. Its inverse sends a relatively open set $V\subset U$ to the basis set ${\hat{V}}_{\omega {|}_V}$, so the inverse is continuous. Conversely the projection of the intersection of ${\hat{U}}_{\omega }$ with any other basis set is an open agreement set like $W$, so the projection is continuous as well. Hence each sheet projects homeomorphically onto its oriented open set. In particular,with both sheets open. This establishes the two-sheeted covering space structure.