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I'm stuck on this problem, I'm trying to prove the following exercise by contrapositive using Zorn's lemma to prove the existence of an atlas-oriented, but I do not succeed, can anyone help me?

Let M be a non-orientable manifold. Prove that exists two charts $(\phi, U)$ and $(\psi, V)$, so that $U,V$ $\subset$ $M$ are connected and $U \cap V$ has two connected components ($Z_1$ and $Z_2$), such that $\det (D( \psi \circ \phi^{-1}))(z_1) > 0 $ and $\det ( D(\psi \circ \phi^{-1}))(z_2) < 0 $ for all $z_1 \in Z_1$, $z_2 \in Z_2$. Where $Df(x)$ is the jacobian of the function $f$ in the point $x$.

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Hint: Find an orientation-reversing loop $c: [0, 1] \to M$. Take a tiny neighborhood of the image of $c$ in $M$. Split it into two pieces: stuff near $c([0, 0.5])$ and stuff near $c([0.5, 1])$. These intersect (if the neighborhood is tiny enough) in a pair of balls that are more or less centered at $c(0)$ and $c(0.5)$. Go from there.

Suggestion: try this with $c$ being the central loop in a Mobius band to get the intuitive idea.

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