Show that the unbounded region determined by a closed curve is doubly connected
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Show that the unbounded region determined by a closed curve is doubly connected.
I am not sure what to do here. I proved that a bounded component determined by the curve is simoly connected, so that I only have to show that the union of such bounded components are connected. But how do I show this? The below proof just assumes this but this is not immediately clear to me.
1
general-topology complex-analysis
asked Nov 8 at 4:11
Cute Brownie
947316
add a comment |
up vote
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down vote
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Show that the unbounded region determined by a closed curve is doubly connected.
I am not sure what to do here. I proved that a bounded component determined by the curve is simoly connected, so that I only have to show that the union of such bounded components are connected. But how do I show this? The below proof just assumes this but this is not immediately clear to me.
1
general-topology complex-analysis
asked Nov 8 at 4:11
Cute Brownie
947316
1
Do you consider a curve in $mathbb{R}^2$? What does "doubly connected" mean in that context?
– Paul Frost
Nov 8 at 12:41
@PaulFrost A doubly connected region $K$ has two components in $mathbb C - K$; for example, the unit circle has two components, one the unit disk and the other the region with infinity.
– Cute Brownie
Nov 15 at 18:15
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Show that the unbounded region determined by a closed curve is doubly connected.
I am not sure what to do here. I proved that a bounded component determined by the curve is simoly connected, so that I only have to show that the union of such bounded components are connected. But how do I show this? The below proof just assumes this but this is not immediately clear to me.
1
general-topology complex-analysis
asked Nov 8 at 4:11
Cute Brownie
947316
Show that the unbounded region determined by a closed curve is doubly connected.
I am not sure what to do here. I proved that a bounded component determined by the curve is simoly connected, so that I only have to show that the union of such bounded components are connected. But how do I show this? The below proof just assumes this but this is not immediately clear to me.
1
general-topology complex-analysis
general-topology complex-analysis
asked Nov 8 at 4:11
Cute Brownie
947316
asked Nov 8 at 4:11
Cute Brownie
947316
edited Nov 8 at 5:19
asked Nov 8 at 4:11
Cute Brownie
947316
asked Nov 8 at 4:11
Cute Brownie
947316
asked Nov 8 at 4:11
Cute Brownie
947316
947316
1
Do you consider a curve in $mathbb{R}^2$? What does "doubly connected" mean in that context?
– Paul Frost
Nov 8 at 12:41
@PaulFrost A doubly connected region $K$ has two components in $mathbb C - K$; for example, the unit circle has two components, one the unit disk and the other the region with infinity.
– Cute Brownie
Nov 15 at 18:15
add a comment |
1
Do you consider a curve in $mathbb{R}^2$? What does "doubly connected" mean in that context?
– Paul Frost
Nov 8 at 12:41
@PaulFrost A doubly connected region $K$ has two components in $mathbb C - K$; for example, the unit circle has two components, one the unit disk and the other the region with infinity.
– Cute Brownie
Nov 15 at 18:15
1
1
Do you consider a curve in $mathbb{R}^2$? What does "doubly connected" mean in that context?
– Paul Frost
Nov 8 at 12:41
Do you consider a curve in $mathbb{R}^2$? What does "doubly connected" mean in that context?
– Paul Frost
Nov 8 at 12:41
@PaulFrost A doubly connected region $K$ has two components in $mathbb C - K$; for example, the unit circle has two components, one the unit disk and the other the region with infinity.
– Cute Brownie
Nov 15 at 18:15
@PaulFrost A doubly connected region $K$ has two components in $mathbb C - K$; for example, the unit circle has two components, one the unit disk and the other the region with infinity.
– Cute Brownie
Nov 15 at 18:15
add a comment |
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1
Do you consider a curve in $mathbb{R}^2$? What does "doubly connected" mean in that context?
– Paul Frost
Nov 8 at 12:41
@PaulFrost A doubly connected region $K$ has two components in $mathbb C - K$; for example, the unit circle has two components, one the unit disk and the other the region with infinity.
– Cute Brownie
Nov 15 at 18:15