= Connectedness criterion for the orientation double cover
{title2=$M\text{ orientable}\iff\widehat M\text{ disconnected}$}
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>.
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