**EDIT:** The well known [Jordan curve theorem](https://en.wikipedia.org/wiki/Jordan_curve_theorem) says: let $C\subset S^2$ be a closed simple curve on the 2-sphere. Then its complement $S^2\backslash C$ consists of two connected components, both homeomorphic to discs (in fact it is known that the closure of each component is homeomorphic to the closed disk by [Jordan-Schoenflies theorem](https://en.wikipedia.org/wiki/Schoenflies_problem)).



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> Is there a version of the Jordan theorem for closed simple curves in real projective plane $\mathbb{R}\mathbb{P}^2$? (The curve might be assumed to be smoothly imbedded.)
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A reference would be helpful.


**ADDED:** Given the comment by HenrikRüping below, I realized that for my purposes it suffices to assume that the homology class of $C$ vanishes in $H\_1(\mathbb{R}\mathbb{P}^2,\mathbb{Z}/2\mathbb{Z})$.



Closed simple curves in $\mathbb{R}\mathbb{P}^2$
If C is null-homologous, then the complement of C has two components: a disk and a Möbius strip (as one sees since the preimage of C in the 2-sphere is 2 disjoint Jordan curves).
If C is not null-homologous, then the complement of C is a single disk.


