Closed simple curves in $\mathbb{R}\mathbb{P}^2$

Question

EDIT: The well known 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).

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.)

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 $H_1(\mathbb{R}\mathbb{P}^2,\mathbb{Z}/2\mathbb{Z})$.

Answer 1

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.

Answer 2

Well, if you take the double cover, under your assumptions the lift is two simple closed curves in $S^2,$ the complement of which will be two disks and an annulus, so the original curve bounds a disk on one side.