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At $(p,o)$ in the orientation double cover, use the derivative of the covering projection to transport the chosen orientation $o$ to the tangent space of the cover. In local oriented sheets this prescription is smooth, so it defines a global orientation. The involution $(p,o)\mapsto (p,-o)$ reverses it.

For a connected smooth manifold, every component of the orientation double cover maps onto the base by the path lifting theorem. There can therefore be at most two components. A global orientation splits the cover into two copies of the base. Conversely, if there are two components, each meets every fibre once, so its inverse projection gives a smooth global choice of orientation.

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.

A connected smooth manifold is path connected because it is locally path connected. Each connected component $C$ of the orientation double cover is itself path connected. Choose a point of $C$ over $p$. For any $q\in M$, lift a path from $p$ to $q$ starting at that chosen point, using the path lifting theorem. The lifted path lies in $C$ and ends above $q$. Therefore every component maps onto $M$.

Fix one fibre, containing exactly two points. Every component meets that fibre, and distinct components meet disjoint subsets of it, so there are at most two connected components.

If $M$ is orientable, a global continuous choice of tangent-space orientation $o$ gives two disjoint sheetsThey are open: on a sufficiently small oriented coordinate neighborhood the global choice agrees consistently with one of the two local sheets. Each projects homeomorphically to the connected base, so these are precisely two connected components.

Conversely suppose the cover is disconnected. It has exactly two components $C_{+},C_{-}$. Both project onto the base and each fibre contains only two points, so each component meets every fibre exactly once. The restriction $\pi :C_{+}\to M$ is consequently a bijective local diffeomorphism, hence a global diffeomorphism. Transporting the canonical orientation of $C_{+}$ through this map or, equivalently, using its smooth inverse section, gives a global orientation of $M$. Thus the connectedness criterion for the orientation double cover is