Curve & Region Terminology Explained

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Discussion Overview

The discussion revolves around the terminology related to curves and regions in the context of vector spaces, focusing on definitions such as smooth, continuous, connected, simply connected, simple, and closed. Participants are examining the correctness of these definitions and their implications in mathematical contexts.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants propose that a curve is smooth if it is infinitely differentiable everywhere in its domain, while others challenge the definition of continuous, asserting it should not involve differentiability.
  • There is contention regarding the definition of connectedness, with some arguing it should be defined in terms of path connectedness and others insisting on a more general definition involving the absence of disconnection.
  • A participant suggests that a region is simply connected if any simple closed curve can be shrunk to a point inside the region, while others question the clarity of this definition.
  • One participant introduces a definition of path connectedness, stating that a region is path connected if any two points can be joined by a curve within the region.
  • Another participant raises a definition of simple regions based on boundaries consisting of horizontal and vertical curves or planes, which is met with confusion and skepticism regarding its clarity and correctness.
  • There is a repeated emphasis on the distinction between connectedness and path connectedness, with examples provided to illustrate that some connected sets are not path connected.
  • Participants express uncertainty about the meaning of "horizontal and vertical curves" in the context of defining simple regions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the definitions discussed. Multiple competing views remain, particularly regarding the definitions of continuous, connected, and simple regions.

Contextual Notes

Definitions are dependent on specific mathematical contexts and may vary based on interpretations. Some definitions provided may lack clarity or precision, leading to confusion among participants.

Swapnil
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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.
 
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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.
 
Last edited:
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.
 
matt grime said:
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.
 
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.
 
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.
 
matt grime said:
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
 
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).
 
Swapnil said:
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.
 

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