Is the given set path connected? A Domain?

[Bashyboy]

Hello everyone,

Here is the definition of path connected and domain my textbook provides:

Definition 171. An open set S is path connected if each pair of points in S can be connected
by a polygonal line (e.g. a finite number of line segments connected end to end). A domain
is an open set that is path connected.

I am asked to determine whether "The set given by ##r>0##, ##- \pi < \Theta < \pi##" is path connected and is a domain.

I just want to make sure I understand exactly what the set is. Is the set they describe ##S = \{(r,\Theta)~|~r >0 \wedge \Theta \in (- \pi, \pi) \}##; and does this set consist of all the points in the complex plane, other than those along the negative real-axis?

If that's the case, it would seem as though it would be path connected.

 

[Dick]

Hello everyone,

Here is the definition of path connected and domain my textbook provides:

Definition 171. An open set S is path connected if each pair of points in S can be connected
by a polygonal line (e.g. a finite number of line segments connected end to end). A domain
is an open set that is path connected.

I am asked to determine whether "The set given by ##r>0##, ##- \pi < \Theta < \pi##" is path connected and is a domain.

I just want to make sure I understand exactly what the set is. Is the set they describe ##S = \{(r,\Theta)~|~r >0 \wedge \Theta \in (- \pi, \pi) \}##; and does this set consist of all the points in the complex plane, other than those along the negative real-axis?

If that's the case, it would seem as though it would be path connected.

Sounds correct so far.

 

[Bashyboy]

I would argue that it is connected because any two points in the plane can be connected by a certain number of polygonal lines, just so long as these lines never pass through the negative real axis.

 

[haruspex]

I would argue that it is connected because any two points in the plane can be connected by a certain number of polygonal lines, just so long as these lines never pass through the negative real axis.

But you have not shown that can be done.

 

[Dick]

I would argue that it is connected because any two points in the plane can be connected by a certain number of polygonal lines, just so long as these lines never pass through the negative real axis.

Fine. That wouldn't be hard to show in detail. Is it a domain?

 

[Bashyboy]

Yes, it is a domain, because the set is an open one, and we have just agreed that it is path connected.

What do you mean by showing in detail? How would you be more detailed in answering this question? I am curious to know.

 

[haruspex]

What do you mean by showing in detail? How would you be more detailed in answering this question?

Not sure, but I think Dick is making the same point I made: your approach is ok but what you have written so far does not prove there exists any such path. Try constructing one: "Given two points in the plane, not lying on the negative X axis or at the origin, with co-ordinates (etc.), the following path contained in the set connects them ..."

 

[Bashyboy]

Oh, I don't believe the book is asking for a rigorous proof, but rather an intuitive consideration of the problem; additionally, I wouldn't really know how to construct a proof because our textbook has not covered it.

 

[haruspex]

Oh, I don't believe the book is asking for a rigorous proof, but rather an intuitive consideration of the problem; additionally, I wouldn't really know how to construct a proof because our textbook has not covered it.

If you say so, but then I wonder what was meant by "determine". Anyway, the construction is pretty easy.

 

[Dick]

Oh, I don't believe the book is asking for a rigorous proof, but rather an intuitive consideration of the problem; additionally, I wouldn't really know how to construct a proof because our textbook has not covered it.

A detailed proof wouldn't be very interesting. But you should still think about how you might make one.