Proof of Jordan-Schoenflies via conformal mappings.
The wikipedia article on Jordan-Schoenflies outlines a proof using the Caratheodory theorem on conformal mappings. My doubt is that the proof uses the Riemann mapping theorem. However we do not apriori know the interior of a Jordan curve is simply connected. I checked Pommerenke's book too. But the confusion persists. Some references would be greatly appreciated.
complex-analysis
asked Nov 28 '18 at 4:27
shc
62
add a comment |
The wikipedia article on Jordan-Schoenflies outlines a proof using the Caratheodory theorem on conformal mappings. My doubt is that the proof uses the Riemann mapping theorem. However we do not apriori know the interior of a Jordan curve is simply connected. I checked Pommerenke's book too. But the confusion persists. Some references would be greatly appreciated.
complex-analysis
asked Nov 28 '18 at 4:27
shc
62
Can't you just use Jordan curve theorem to get the interior being simply connected? Or are you trying to prove Jordan curve theorem by Jordan-Schoenflies?
– user10354138
Nov 28 '18 at 4:42
I don't see how.
– shc
Nov 28 '18 at 4:45
Even if we assume the Jordan Curve Theorem how do we get the interior is simply connected from it? Could you elaborate?
– shc
Nov 28 '18 at 4:50
From JCT, there is an alternative characterisation of simply-connectedness in $mathbb{C}$: an open subset $Omegasubseteqmathbb{C}$ is simply-connected iff for every Jordan curve in $Omega$ the interior is contained in $Omega$. Then it follows immediately that interior of a Jordan curve must be simply-connected.
– user10354138
Nov 30 '18 at 12:19
add a comment |
The wikipedia article on Jordan-Schoenflies outlines a proof using the Caratheodory theorem on conformal mappings. My doubt is that the proof uses the Riemann mapping theorem. However we do not apriori know the interior of a Jordan curve is simply connected. I checked Pommerenke's book too. But the confusion persists. Some references would be greatly appreciated.
complex-analysis
asked Nov 28 '18 at 4:27
shc
62
The wikipedia article on Jordan-Schoenflies outlines a proof using the Caratheodory theorem on conformal mappings. My doubt is that the proof uses the Riemann mapping theorem. However we do not apriori know the interior of a Jordan curve is simply connected. I checked Pommerenke's book too. But the confusion persists. Some references would be greatly appreciated.
complex-analysis
complex-analysis
asked Nov 28 '18 at 4:27
shc
62
asked Nov 28 '18 at 4:27
shc
62
asked Nov 28 '18 at 4:27
shc
62
asked Nov 28 '18 at 4:27
shc
62
asked Nov 28 '18 at 4:27
shc
62
62
Can't you just use Jordan curve theorem to get the interior being simply connected? Or are you trying to prove Jordan curve theorem by Jordan-Schoenflies?
– user10354138
Nov 28 '18 at 4:42
I don't see how.
– shc
Nov 28 '18 at 4:45
Even if we assume the Jordan Curve Theorem how do we get the interior is simply connected from it? Could you elaborate?
– shc
Nov 28 '18 at 4:50
From JCT, there is an alternative characterisation of simply-connectedness in $mathbb{C}$: an open subset $Omegasubseteqmathbb{C}$ is simply-connected iff for every Jordan curve in $Omega$ the interior is contained in $Omega$. Then it follows immediately that interior of a Jordan curve must be simply-connected.
– user10354138
Nov 30 '18 at 12:19
add a comment |
Can't you just use Jordan curve theorem to get the interior being simply connected? Or are you trying to prove Jordan curve theorem by Jordan-Schoenflies?
– user10354138
Nov 28 '18 at 4:42
I don't see how.
– shc
Nov 28 '18 at 4:45
Even if we assume the Jordan Curve Theorem how do we get the interior is simply connected from it? Could you elaborate?
– shc
Nov 28 '18 at 4:50
From JCT, there is an alternative characterisation of simply-connectedness in $mathbb{C}$: an open subset $Omegasubseteqmathbb{C}$ is simply-connected iff for every Jordan curve in $Omega$ the interior is contained in $Omega$. Then it follows immediately that interior of a Jordan curve must be simply-connected.
– user10354138
Nov 30 '18 at 12:19
Can't you just use Jordan curve theorem to get the interior being simply connected? Or are you trying to prove Jordan curve theorem by Jordan-Schoenflies?
– user10354138
Nov 28 '18 at 4:42
Can't you just use Jordan curve theorem to get the interior being simply connected? Or are you trying to prove Jordan curve theorem by Jordan-Schoenflies?
– user10354138
Nov 28 '18 at 4:42
I don't see how.
– shc
Nov 28 '18 at 4:45
I don't see how.
– shc
Nov 28 '18 at 4:45
Even if we assume the Jordan Curve Theorem how do we get the interior is simply connected from it? Could you elaborate?
– shc
Nov 28 '18 at 4:50
Even if we assume the Jordan Curve Theorem how do we get the interior is simply connected from it? Could you elaborate?
– shc
Nov 28 '18 at 4:50
From JCT, there is an alternative characterisation of simply-connectedness in $mathbb{C}$: an open subset $Omegasubseteqmathbb{C}$ is simply-connected iff for every Jordan curve in $Omega$ the interior is contained in $Omega$. Then it follows immediately that interior of a Jordan curve must be simply-connected.
– user10354138
Nov 30 '18 at 12:19
From JCT, there is an alternative characterisation of simply-connectedness in $mathbb{C}$: an open subset $Omegasubseteqmathbb{C}$ is simply-connected iff for every Jordan curve in $Omega$ the interior is contained in $Omega$. Then it follows immediately that interior of a Jordan curve must be simply-connected.
– user10354138
Nov 30 '18 at 12:19
add a comment |
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Can't you just use Jordan curve theorem to get the interior being simply connected? Or are you trying to prove Jordan curve theorem by Jordan-Schoenflies?
– user10354138
Nov 28 '18 at 4:42
I don't see how.
– shc
Nov 28 '18 at 4:45
Even if we assume the Jordan Curve Theorem how do we get the interior is simply connected from it? Could you elaborate?
– shc
Nov 28 '18 at 4:50
From JCT, there is an alternative characterisation of simply-connectedness in $mathbb{C}$: an open subset $Omegasubseteqmathbb{C}$ is simply-connected iff for every Jordan curve in $Omega$ the interior is contained in $Omega$. Then it follows immediately that interior of a Jordan curve must be simply-connected.
– user10354138
Nov 30 '18 at 12:19