Simply connected Region

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A region in the complex plane is said to be simply connected if any simple closed curve in the region can be shrunk or continuously deformed to a point in the region.

My question is: How can i understand the intuitive meaning of this definition without using the fact that the simply connected region has no hole in it? Please what does it mean that the simple closed curve is continuously deformed to a point?
 

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  • #2
HallsofIvy
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A region in the complex plane is said to be simply connected if any simple closed curve in the region can be shrunk or continuously deformed to a point in the region.

My question is: How can i understand the intuitive meaning of this definition without using the fact that the simply connected region has no hole in it? Please what does it mean that the simple closed curve is continuously deformed to a point?
I'm not sure what you mean by "without using the fact that the simply connected region has no hole in it". That is the "intuitive meaning"! If there is a "hole" in a region in the plane, then the boundary of that hole cannot be shrunk to a point.
 
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I'm not sure what you mean by "without using the fact that the simply connected region has no hole in it". That is the "intuitive meaning"! If there is a "hole" in a region in the plane, then the boundary of that hole cannot be shrunk to a point.
Are you saying the hole is a is a closed curve in the region? ofcourse yes. If so how can a closed curve be continuously deformed or shrunk to a point in the region
 
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Imagine your curve is a piece of string with a little loop at one end and the other end going through the loop, so that the string is lying fully inside your region forming a closed curve - now slowly pull the non-looped end through the loop so that the closed curve shrinks within your region. If you can shrink the curve to a point without it leaving the region, and you can always do so no matter how you lay your thread down initially as long as it lies within the region, then your region is simply connected - if on the other hand, as you shrink the curve there comes a point when it must cross the region's boundary (because there is a hole contained within), then your region is not simply connected. You can try using a string on a piece of cardboard and put some obstacle in the middle representing the hole :P if your string initially surrounds the obstacle, then as you shrink it it will end up wrapping itself around the obstacle and you won't be able to shrink it any further.
 
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Please what does it mean that the simple closed curve is continuously deformed to a point?

The deformation is called a homotopy and it is a function (map) that sort of parametrizes possible continuous loops using a parameter from 0 to 1.

For example, if we wanted to deform an ellipse to a circle of radius one, F is the homotopy, and at parameter 0 it gives a function f that describes an ellipse.

we plot the ellipse in a plane, let's say F(0) = f: {x = 3 cos(t), y = 5 sin(t)}

once we run the parameter from 0 to 1, F will spit out different loops between the ellipse and the circle, until finally at 1 we get a circle,

F(1) = g: {x = cos(t), y = sin(t)}


We can also deform the ellipse to a single point, the homotopy F will spit out all loops between ellipse and a point. And if there is a hole in space, the continuity will brake.

There is a better way to describe this using fundamental groups, but it's more advanced.
 
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