Showing a subset of $mathbb{C}$ is simply connected via homotopy
$begingroup$
This question is Exercise 19 in Chpater 8 of Stein and Shakarchi's Complex Analysis.
Prove that the complex plane slit along the union of rays $cup_{k=1}^{n}{A_k+iy:yleq 0}, A_k in mathbb{R}$ is simply connected.
This question asks for clarification about this problem, which was helpful but I am still having trouble understanding how to go about proving this claim. There is an answer given in the previously mentioned question but uses the notion of a deformation retract which I am not familiar with.
Ideally, I would like to prove this claim by showing that any two curves in our region with the same end points are homotopic. The book gives the hint
Hint: Given a curve first "raise" it so it is completely contained in the upper half-plane.
while my professor hinted that we should first prove that homotopies are transitive.
So my current thoughts are that we are raising our curves so that we can use (or prove) the fact that the upper-half plane is simply connected and thus any two curves with the same end points are homotopic. What I am having trouble visualizing is how we explicitly include the ray-slits in the proof.
If someone could orient me on the right path I would be very appreciative!
complex-analysis connectedness
asked Dec 5 '18 at 3:39
FofXFofX
637416
$endgroup$
add a comment |
$begingroup$
This question is Exercise 19 in Chpater 8 of Stein and Shakarchi's Complex Analysis.
Prove that the complex plane slit along the union of rays $cup_{k=1}^{n}{A_k+iy:yleq 0}, A_k in mathbb{R}$ is simply connected.
This question asks for clarification about this problem, which was helpful but I am still having trouble understanding how to go about proving this claim. There is an answer given in the previously mentioned question but uses the notion of a deformation retract which I am not familiar with.
Ideally, I would like to prove this claim by showing that any two curves in our region with the same end points are homotopic. The book gives the hint
Hint: Given a curve first "raise" it so it is completely contained in the upper half-plane.
while my professor hinted that we should first prove that homotopies are transitive.
So my current thoughts are that we are raising our curves so that we can use (or prove) the fact that the upper-half plane is simply connected and thus any two curves with the same end points are homotopic. What I am having trouble visualizing is how we explicitly include the ray-slits in the proof.
If someone could orient me on the right path I would be very appreciative!
complex-analysis connectedness
asked Dec 5 '18 at 3:39
FofXFofX
637416
$endgroup$
add a comment |
$begingroup$
This question is Exercise 19 in Chpater 8 of Stein and Shakarchi's Complex Analysis.
Prove that the complex plane slit along the union of rays $cup_{k=1}^{n}{A_k+iy:yleq 0}, A_k in mathbb{R}$ is simply connected.
This question asks for clarification about this problem, which was helpful but I am still having trouble understanding how to go about proving this claim. There is an answer given in the previously mentioned question but uses the notion of a deformation retract which I am not familiar with.
Ideally, I would like to prove this claim by showing that any two curves in our region with the same end points are homotopic. The book gives the hint
Hint: Given a curve first "raise" it so it is completely contained in the upper half-plane.
while my professor hinted that we should first prove that homotopies are transitive.
So my current thoughts are that we are raising our curves so that we can use (or prove) the fact that the upper-half plane is simply connected and thus any two curves with the same end points are homotopic. What I am having trouble visualizing is how we explicitly include the ray-slits in the proof.
If someone could orient me on the right path I would be very appreciative!
complex-analysis connectedness
asked Dec 5 '18 at 3:39
FofXFofX
637416
$endgroup$
This question is Exercise 19 in Chpater 8 of Stein and Shakarchi's Complex Analysis.
Prove that the complex plane slit along the union of rays $cup_{k=1}^{n}{A_k+iy:yleq 0}, A_k in mathbb{R}$ is simply connected.
This question asks for clarification about this problem, which was helpful but I am still having trouble understanding how to go about proving this claim. There is an answer given in the previously mentioned question but uses the notion of a deformation retract which I am not familiar with.
Ideally, I would like to prove this claim by showing that any two curves in our region with the same end points are homotopic. The book gives the hint
Hint: Given a curve first "raise" it so it is completely contained in the upper half-plane.
while my professor hinted that we should first prove that homotopies are transitive.
So my current thoughts are that we are raising our curves so that we can use (or prove) the fact that the upper-half plane is simply connected and thus any two curves with the same end points are homotopic. What I am having trouble visualizing is how we explicitly include the ray-slits in the proof.
If someone could orient me on the right path I would be very appreciative!
complex-analysis connectedness
complex-analysis connectedness
asked Dec 5 '18 at 3:39
FofXFofX
637416
asked Dec 5 '18 at 3:39
FofXFofX
637416
asked Dec 5 '18 at 3:39
FofXFofX
637416
asked Dec 5 '18 at 3:39
FofXFofX
637416
asked Dec 5 '18 at 3:39
FofXFofX
637416
637416
add a comment |
add a comment |
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