What is the intuitive meaning of a simply connected region?

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In summary, a region in the complex plane is simply connected if any simple closed curve within the region can be continuously deformed to a point without leaving the region. This means that there are no holes in the region, as the curve can always be shrunk to a point without crossing the boundary. This is the intuitive meaning of the definition and can be visualized by thinking of a piece of string being shrunk within the region. The process of continuous deformation is called a homotopy and is represented by a function that parametrizes the possible loops between the curve and a point or another curve. This concept can also be described using fundamental groups, but it is more advanced.
  • #1
de_brook
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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
de_brook said:
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.
 
  • #3
HallsofIvy said:
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
 
  • #4
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.
 
  • #5
de_brook said:
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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1. What is a simply connected region?

A simply connected region is a type of mathematical concept used in topology. It refers to a region or space that is connected and does not have any "holes" or "handles". This means that any loop drawn in the region can be continuously shrunk to a single point without leaving the region.

2. How is a simply connected region different from a connected region?

A simply connected region is a type of connected region, but it has the additional requirement that there are no holes or handles. A connected region may have multiple components or holes, while a simply connected region does not.

3. Can a simply connected region have a boundary?

Yes, a simply connected region can have a boundary. The boundary is considered to be part of the region and does not affect its simply connected property. This means that any loop drawn in the region can still be continuously shrunk to a single point without leaving the region, even if the loop crosses the boundary.

4. What is the importance of simply connected regions in mathematics?

Simply connected regions are important in many areas of mathematics, including topology, complex analysis, and differential geometry. They provide a way to classify and study different types of spaces and can aid in solving problems related to these areas.

5. How can you determine if a region is simply connected?

There are a few different methods for determining if a region is simply connected. One way is to check if any loop can be continuously shrunk to a single point without leaving the region. Another way is to use algebraic topology, which involves looking at the fundamental group of the region. If the fundamental group is trivial (meaning it only contains the identity element), then the region is simply connected.

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