Prove that a star-shaped open set is simply connected with diffierent definition of simple connected .
$begingroup$
Prove that a star-shaped open set is simply connected.
This is an exercise in Stein's complex analysis and I have already seen one of the proof here.
In Stein's book , A region $Omega$ in the complex plane is simply connected if any two pair of curves in $Omega$ with the same end-points are homotopic. But in the proof above , it seems to apply another definition of simply connected (any closed curve in the region is homotopic to a point in the region?) .
My question:
a) How to prove this exercise with the definition of simple connected in Stein's book .
b) What kind of the definition of simple connected did the proof above applied . Are the two definition of simple connected the same ? If so , how to show this ?
complex-analysis
asked Jan 5 at 7:04
J.GuoJ.Guo
3569
$endgroup$
add a comment |
$begingroup$
Prove that a star-shaped open set is simply connected.
This is an exercise in Stein's complex analysis and I have already seen one of the proof here.
In Stein's book , A region $Omega$ in the complex plane is simply connected if any two pair of curves in $Omega$ with the same end-points are homotopic. But in the proof above , it seems to apply another definition of simply connected (any closed curve in the region is homotopic to a point in the region?) .
My question:
a) How to prove this exercise with the definition of simple connected in Stein's book .
b) What kind of the definition of simple connected did the proof above applied . Are the two definition of simple connected the same ? If so , how to show this ?
complex-analysis
asked Jan 5 at 7:04
J.GuoJ.Guo
3569
$endgroup$
add a comment |
$begingroup$
Prove that a star-shaped open set is simply connected.
This is an exercise in Stein's complex analysis and I have already seen one of the proof here.
In Stein's book , A region $Omega$ in the complex plane is simply connected if any two pair of curves in $Omega$ with the same end-points are homotopic. But in the proof above , it seems to apply another definition of simply connected (any closed curve in the region is homotopic to a point in the region?) .
My question:
a) How to prove this exercise with the definition of simple connected in Stein's book .
b) What kind of the definition of simple connected did the proof above applied . Are the two definition of simple connected the same ? If so , how to show this ?
complex-analysis
asked Jan 5 at 7:04
J.GuoJ.Guo
3569
$endgroup$
Prove that a star-shaped open set is simply connected.
This is an exercise in Stein's complex analysis and I have already seen one of the proof here.
In Stein's book , A region $Omega$ in the complex plane is simply connected if any two pair of curves in $Omega$ with the same end-points are homotopic. But in the proof above , it seems to apply another definition of simply connected (any closed curve in the region is homotopic to a point in the region?) .
My question:
a) How to prove this exercise with the definition of simple connected in Stein's book .
b) What kind of the definition of simple connected did the proof above applied . Are the two definition of simple connected the same ? If so , how to show this ?
complex-analysis
complex-analysis
asked Jan 5 at 7:04
J.GuoJ.Guo
3569
asked Jan 5 at 7:04
J.GuoJ.Guo
3569
asked Jan 5 at 7:04
J.GuoJ.Guo
3569
asked Jan 5 at 7:04
J.GuoJ.Guo
3569
asked Jan 5 at 7:04
J.GuoJ.Guo
3569
3569
add a comment |
add a comment |
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