= Orientable total space of the orientation line bundle
The total space of the <orientation line bundle> is orientable even when its base is not: an orientation reversal in a base coordinate transition is matched by a reversal in the line coordinate, so their product has positive sign. Fiberwise multiplication by $1-t$ gives a <deformation retraction> onto the zero section. For the real projective plane this total space is $(S^2\times\mathbb R)/((v,t)\sim(-v,-t))$, an oriented noncompact three-manifold homotopy equivalent to the plane.
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