We say that the complement of a strip a≤x≤b is connected in the extended complex plane (or Riemann Sphere) and disconnected in a complex plane. Why? [closed]

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My another doubt is how can a point, infinity connect the two disconnected parts in the Riemann sphere?

complex-analysis complex-geometry riemann-sphere

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edited Jul 28 at 18:54

asked Jul 28 at 18:22

Abraham Dinesh

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closed as unclear what you're asking by user 108128, amWhy, Mostafa Ayaz, John Ma, John B Jul 28 at 20:35

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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  Do you know stereographic projection?
  – Randall
  Jul 28 at 18:23
  

  
  
  
  

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My another doubt is how can a point, infinity connect the two disconnected parts in the Riemann sphere?

complex-analysis complex-geometry riemann-sphere

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edited Jul 28 at 18:54

asked Jul 28 at 18:22

Abraham Dinesh

41

closed as unclear what you're asking by user 108128, amWhy, Mostafa Ayaz, John Ma, John B Jul 28 at 20:35

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Do you know stereographic projection?
  – Randall
  Jul 28 at 18:23
  

  
  
  
  

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

My another doubt is how can a point, infinity connect the two disconnected parts in the Riemann sphere?

complex-analysis complex-geometry riemann-sphere

share|cite|improve this question

edited Jul 28 at 18:54

asked Jul 28 at 18:22

Abraham Dinesh

41

My another doubt is how can a point, infinity connect the two disconnected parts in the Riemann sphere?

complex-analysis complex-geometry riemann-sphere

share|cite|improve this question

edited Jul 28 at 18:54

asked Jul 28 at 18:22

Abraham Dinesh

41

share|cite|improve this question

share|cite|improve this question

edited Jul 28 at 18:54

edited Jul 28 at 18:54

edited Jul 28 at 18:54

asked Jul 28 at 18:22

Abraham Dinesh

41

asked Jul 28 at 18:22

Abraham Dinesh

41

asked Jul 28 at 18:22

Abraham Dinesh

41

41

closed as unclear what you're asking by user 108128, amWhy, Mostafa Ayaz, John Ma, John B Jul 28 at 20:35

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

closed as unclear what you're asking by user 108128, amWhy, Mostafa Ayaz, John Ma, John B Jul 28 at 20:35

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Do you know stereographic projection?
  – Randall
  Jul 28 at 18:23
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Do you know stereographic projection?
  – Randall
  Jul 28 at 18:23
  

  
  
  
  

Do you know stereographic projection?
– Randall
Jul 28 at 18:23

Do you know stereographic projection?
– Randall
Jul 28 at 18:23

add a comment | 

2 Answers
2

active

oldest

votes

up vote
1
down vote

Because in the Riemann sphere, we can join any two points of that set using a path (eventually going through $infty$), whereas in $mathbb C$ that can't be done if one of the points is such that its real part is smaller than $a$ and the other one is such that its real part is greater than $b$.

share|cite|improve this answer

answered Jul 28 at 18:26

José Carlos Santos

112k1696173

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thank you sir. But my doubt is that when any strip a≤y≤b disconnects the sphere into upper and lower hemispheres how will we connect a point in the lower hemisphere to a point in upper hemisphere. Because infinity itself is in the upper hemisphere
  – Abraham Dinesh
  Jul 28 at 18:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Your strip is vertical? Or is it horizontal?
  – José Carlos Santos
  Jul 28 at 18:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Horizontal strip
  – Abraham Dinesh
  Jul 28 at 18:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I thought that it was vertical, but that doesn't really mater, because in any case the complement of the strip is connected in the Riemann sphere. In the Riemann, the stripo is bounded by two circles (not great circles), both of which pass through the point at infinity.
  – José Carlos Santos
  Jul 28 at 18:50
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes, in the Riemann Sphere the complement of the strip is connected. But how? When it looks like its disconnected how do we say its connected? The reason behind it.
  – Abraham Dinesh
  Jul 28 at 18:58
  

  
  
  
  

 | 
show 1 more comment

up vote
1
down vote

You ever cut an orange into wedges? Put the orange back together but leave out ONE of the wedges. The surface of the peel is what you get in the Riemann sphere after you delete out your infinite strip. The orange skin is still connected. The "ends" of the infinite strip have been tied together at the "top" of the orange.

In the plane, this is instead like cutting out a strip from the middle of a sheet of paper, which then separates the remaining paper into two pieces.

share|cite|improve this answer

answered Jul 28 at 19:06

Randall

7,2221825

add a comment | 

2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
1
down vote

Because in the Riemann sphere, we can join any two points of that set using a path (eventually going through $infty$), whereas in $mathbb C$ that can't be done if one of the points is such that its real part is smaller than $a$ and the other one is such that its real part is greater than $b$.

share|cite|improve this answer

answered Jul 28 at 18:26

José Carlos Santos

112k1696173

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thank you sir. But my doubt is that when any strip a≤y≤b disconnects the sphere into upper and lower hemispheres how will we connect a point in the lower hemisphere to a point in upper hemisphere. Because infinity itself is in the upper hemisphere
  – Abraham Dinesh
  Jul 28 at 18:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Your strip is vertical? Or is it horizontal?
  – José Carlos Santos
  Jul 28 at 18:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Horizontal strip
  – Abraham Dinesh
  Jul 28 at 18:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I thought that it was vertical, but that doesn't really mater, because in any case the complement of the strip is connected in the Riemann sphere. In the Riemann, the stripo is bounded by two circles (not great circles), both of which pass through the point at infinity.
  – José Carlos Santos
  Jul 28 at 18:50
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes, in the Riemann Sphere the complement of the strip is connected. But how? When it looks like its disconnected how do we say its connected? The reason behind it.
  – Abraham Dinesh
  Jul 28 at 18:58
  

  
  
  
  

 | 
show 1 more comment

up vote
1
down vote

Because in the Riemann sphere, we can join any two points of that set using a path (eventually going through $infty$), whereas in $mathbb C$ that can't be done if one of the points is such that its real part is smaller than $a$ and the other one is such that its real part is greater than $b$.

share|cite|improve this answer

answered Jul 28 at 18:26

José Carlos Santos

112k1696173

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thank you sir. But my doubt is that when any strip a≤y≤b disconnects the sphere into upper and lower hemispheres how will we connect a point in the lower hemisphere to a point in upper hemisphere. Because infinity itself is in the upper hemisphere
  – Abraham Dinesh
  Jul 28 at 18:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Your strip is vertical? Or is it horizontal?
  – José Carlos Santos
  Jul 28 at 18:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Horizontal strip
  – Abraham Dinesh
  Jul 28 at 18:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I thought that it was vertical, but that doesn't really mater, because in any case the complement of the strip is connected in the Riemann sphere. In the Riemann, the stripo is bounded by two circles (not great circles), both of which pass through the point at infinity.
  – José Carlos Santos
  Jul 28 at 18:50
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes, in the Riemann Sphere the complement of the strip is connected. But how? When it looks like its disconnected how do we say its connected? The reason behind it.
  – Abraham Dinesh
  Jul 28 at 18:58
  

  
  
  
  

 | 
show 1 more comment

up vote
1
down vote

up vote
1
down vote

Because in the Riemann sphere, we can join any two points of that set using a path (eventually going through $infty$), whereas in $mathbb C$ that can't be done if one of the points is such that its real part is smaller than $a$ and the other one is such that its real part is greater than $b$.

share|cite|improve this answer

answered Jul 28 at 18:26

José Carlos Santos

112k1696173

Because in the Riemann sphere, we can join any two points of that set using a path (eventually going through $infty$), whereas in $mathbb C$ that can't be done if one of the points is such that its real part is smaller than $a$ and the other one is such that its real part is greater than $b$.

share|cite|improve this answer

answered Jul 28 at 18:26

José Carlos Santos

112k1696173

share|cite|improve this answer

share|cite|improve this answer

answered Jul 28 at 18:26

José Carlos Santos

112k1696173

answered Jul 28 at 18:26

José Carlos Santos

112k1696173

answered Jul 28 at 18:26

José Carlos Santos

112k1696173

112k1696173

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thank you sir. But my doubt is that when any strip a≤y≤b disconnects the sphere into upper and lower hemispheres how will we connect a point in the lower hemisphere to a point in upper hemisphere. Because infinity itself is in the upper hemisphere
  – Abraham Dinesh
  Jul 28 at 18:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Your strip is vertical? Or is it horizontal?
  – José Carlos Santos
  Jul 28 at 18:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Horizontal strip
  – Abraham Dinesh
  Jul 28 at 18:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I thought that it was vertical, but that doesn't really mater, because in any case the complement of the strip is connected in the Riemann sphere. In the Riemann, the stripo is bounded by two circles (not great circles), both of which pass through the point at infinity.
  – José Carlos Santos
  Jul 28 at 18:50
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes, in the Riemann Sphere the complement of the strip is connected. But how? When it looks like its disconnected how do we say its connected? The reason behind it.
  – Abraham Dinesh
  Jul 28 at 18:58
  

  
  
  
  

 | 
show 1 more comment

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thank you sir. But my doubt is that when any strip a≤y≤b disconnects the sphere into upper and lower hemispheres how will we connect a point in the lower hemisphere to a point in upper hemisphere. Because infinity itself is in the upper hemisphere
  – Abraham Dinesh
  Jul 28 at 18:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Your strip is vertical? Or is it horizontal?
  – José Carlos Santos
  Jul 28 at 18:40
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Horizontal strip
  – Abraham Dinesh
  Jul 28 at 18:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I thought that it was vertical, but that doesn't really mater, because in any case the complement of the strip is connected in the Riemann sphere. In the Riemann, the stripo is bounded by two circles (not great circles), both of which pass through the point at infinity.
  – José Carlos Santos
  Jul 28 at 18:50
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes, in the Riemann Sphere the complement of the strip is connected. But how? When it looks like its disconnected how do we say its connected? The reason behind it.
  – Abraham Dinesh
  Jul 28 at 18:58
  

  
  
  
  

Thank you sir. But my doubt is that when any strip a≤y≤b disconnects the sphere into upper and lower hemispheres how will we connect a point in the lower hemisphere to a point in upper hemisphere. Because infinity itself is in the upper hemisphere
– Abraham Dinesh
Jul 28 at 18:37

Thank you sir. But my doubt is that when any strip a≤y≤b disconnects the sphere into upper and lower hemispheres how will we connect a point in the lower hemisphere to a point in upper hemisphere. Because infinity itself is in the upper hemisphere
– Abraham Dinesh
Jul 28 at 18:37

Your strip is vertical? Or is it horizontal?
– José Carlos Santos
Jul 28 at 18:40

Your strip is vertical? Or is it horizontal?
– José Carlos Santos
Jul 28 at 18:40

Horizontal strip
– Abraham Dinesh
Jul 28 at 18:41

Horizontal strip
– Abraham Dinesh
Jul 28 at 18:41

I thought that it was vertical, but that doesn't really mater, because in any case the complement of the strip is connected in the Riemann sphere. In the Riemann, the stripo is bounded by two circles (not great circles), both of which pass through the point at infinity.
– José Carlos Santos
Jul 28 at 18:50

I thought that it was vertical, but that doesn't really mater, because in any case the complement of the strip is connected in the Riemann sphere. In the Riemann, the stripo is bounded by two circles (not great circles), both of which pass through the point at infinity.
– José Carlos Santos
Jul 28 at 18:50

Yes, in the Riemann Sphere the complement of the strip is connected. But how? When it looks like its disconnected how do we say its connected? The reason behind it.
– Abraham Dinesh
Jul 28 at 18:58

Yes, in the Riemann Sphere the complement of the strip is connected. But how? When it looks like its disconnected how do we say its connected? The reason behind it.
– Abraham Dinesh
Jul 28 at 18:58

 | 
show 1 more comment

up vote
1
down vote

You ever cut an orange into wedges? Put the orange back together but leave out ONE of the wedges. The surface of the peel is what you get in the Riemann sphere after you delete out your infinite strip. The orange skin is still connected. The "ends" of the infinite strip have been tied together at the "top" of the orange.

In the plane, this is instead like cutting out a strip from the middle of a sheet of paper, which then separates the remaining paper into two pieces.

share|cite|improve this answer

answered Jul 28 at 19:06

Randall

7,2221825

add a comment | 

up vote
1
down vote

You ever cut an orange into wedges? Put the orange back together but leave out ONE of the wedges. The surface of the peel is what you get in the Riemann sphere after you delete out your infinite strip. The orange skin is still connected. The "ends" of the infinite strip have been tied together at the "top" of the orange.

In the plane, this is instead like cutting out a strip from the middle of a sheet of paper, which then separates the remaining paper into two pieces.

share|cite|improve this answer

answered Jul 28 at 19:06

Randall

7,2221825

add a comment | 

up vote
1
down vote

up vote
1
down vote

You ever cut an orange into wedges? Put the orange back together but leave out ONE of the wedges. The surface of the peel is what you get in the Riemann sphere after you delete out your infinite strip. The orange skin is still connected. The "ends" of the infinite strip have been tied together at the "top" of the orange.

In the plane, this is instead like cutting out a strip from the middle of a sheet of paper, which then separates the remaining paper into two pieces.

share|cite|improve this answer

answered Jul 28 at 19:06

Randall

7,2221825

You ever cut an orange into wedges? Put the orange back together but leave out ONE of the wedges. The surface of the peel is what you get in the Riemann sphere after you delete out your infinite strip. The orange skin is still connected. The "ends" of the infinite strip have been tied together at the "top" of the orange.

In the plane, this is instead like cutting out a strip from the middle of a sheet of paper, which then separates the remaining paper into two pieces.

share|cite|improve this answer

answered Jul 28 at 19:06

Randall

7,2221825

share|cite|improve this answer

share|cite|improve this answer

answered Jul 28 at 19:06

Randall

7,2221825

answered Jul 28 at 19:06

Randall

7,2221825

answered Jul 28 at 19:06

Randall

7,2221825

7,2221825

add a comment | 

add a comment |