Let's say you divide the real projective plane into two subsets, are exactly one these subsets non-orientable?
In particular, we will require that each subset $S$ is "nice" in the sense their common boundary consists of a finite number of non-intersecting polygons (in particular, this excludes 1 dimensional regions).
Note the answer when divide it into three subsets is "no". You can divide the real projective plane into three orientable components.
(The motivation for this question is that, if the answer to this question is yes, you can play a variant of the Hex board game where you win if you form a non-orientable subset of the projective plane.)
EDIT: There actually is a game (called Projex) based on this concept. The website seems to imply that the answer to this question is yes. It is based on a discrete version of this problem though, and includes no proof.