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

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.

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

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asv
asv
  • 21.8k
  • 6
  • 54
  • 121

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.)

My guess would be that the complement also has exactly two connected components: one homeomorphisc to disc, and the other one to the Moebius band, but I am not sure. A reference would be helpful.

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.)

My guess would be that the complement also has exactly two connected components: one homeomorphisc to disc, and the other one to the Moebius band, but I am not sure. A reference would be helpful.

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.

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asv
asv
  • 21.8k
  • 6
  • 54
  • 121

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

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.)

My guess would be that the complement also has exactly two connected components: one homeomorphisc to disc, and the other one to the Moebius band, but I am not sure. A reference would be helpful.