Curve & Region Terminology Explained

[Swapnil]

I am a little confused about some terminology when we talk about curves and regions (in the context of vectors for example).

Are these correct?

Curve Terminology:

Smooth: A curve C is said to be smooth if it is infinitely differentiable everywhere in its domain.

Continuous: A curve C is said to be continuous if it is once differentiable everywhere in its domain.

Simple: A curve C is said to be simple if it does not cross itself.

Region Terminology:

Connected: A region D in R^3 is said to be connected if any two points in D can be joined by a smooth curve completely in D.

Simply Connected: A region D in R^3, is said to be be simply connected if any simple closed curve inside D can be shrunk down to a point inside D.

 

[matt grime]

Continuous is wrong. Continuous just means continuous. Nothing to do with differentiable.

Connected is also wrong. It is connected if it is not disconnected. A space is disconnected if there exist open sets X and Y such that D=XuY, and XnY is empty.

If you omit the word 'smooth' you have described path connected.

These are different: the space { (x,sin(1/x)), 0<x<1} u { (0,y) : -1<y<1} is connected but not path connected.

 

[Gib Z]

A curve in only differentiable if it is continuous, you can't use that in your definition for Continuous. matt grime is correct, it means what a layman would think it means.

 

[Swapnil]

Continuous is wrong. Continuous just means continuous. Nothing to do with differentiable.

Connected is also wrong. It is connected if it is not disconnected. A space is disconnected if there exist open sets X and Y such that D=XuY, and XnY is empty.

If you omit the word 'smooth' you have described path connected.

Thanks. So ...
Path Connected: A region D in R^3 is said to be path connected if any two points in D can be joined by a curve completely in D.

Also, is this one correct?

Simple: A region D in R^2 is called simple if its boundary consists of only horizontal and vertical curves. A region D in R^3 is simple if its boundary consists of only horizontal and vertical planes.

 

[HallsofIvy]

By the way, it's faily easy to show that any path connected set is connected, that any open, connected set is path connected and that any closed, connected set is path connected. However, there exist connected sets that are not path connected.

 

[Swapnil]

So are these correct or not:?

Simple: A region D in R^2 is called simple if its boundary consists of only horizontal and vertical curves. A region D in R^3 is simple if its boundary consists of only horizontal and vertical planes.

Closed: A curve C is said to be closed if its endpoints are the same.

 

[Swapnil]

Connected is also wrong. It is connected if it is not disconnected. A space is disconnected if there exist open sets X and Y such that D=XuY, and XnY is empty.

What about this one?
Connected: A region D is called connected if for any two points P and Q in D, there is a curve C with endpoints P and Q.

I found this definition here:
http://www.math.utah.edu/online/2210/notes/ch18.pdf

 

[matt grime]

We have already told you the deinfition of connected, and demonstrated that it is not equivalent to the condition you give which we also explained was what path connected meant (assuming you mean C to lie in D - if you don't it is a peculiar definition).

 

[HallsofIvy]

So are these correct or not:?

Simple: A region D in R^2 is called simple if its boundary consists of only horizontal and vertical curves. A region D in R^3 is simple if its boundary consists of only horizontal and vertical planes.

Closed: A curve C is said to be closed if its endpoints are the same.

I have no idea what "horizontal and vertical curves" could MEAN! A region is "simple" if its boundary is connected.