Complex Made Simple: Notation on Disks

[BrainHurts]

Hello,

I'm reading "Complex Made Simple" by David Ullrich. He has these notation for disks

[itex] D(z_0,r) = \left\{ z \in \mathbb{C}: |z-z_0|< r \right\} [/itex]

[itex] \bar{D}(z_0,r) = \left\{ z \in \mathbb{C} : |z - z_0| \leq r \right\}[/itex]

I understand that these sets are to be the open and closed disks with radius r respectively.

The one I'm not sure about is what does [itex] \overline{D(z_0,r)} [/itex] mean? Any thoughts?

 

[Office_Shredder]

That means the topological closure of the set [itex] D(z_0, r) [/itex]. It turns out to be equal to [itex] \overline{D}(z_0, r)[/itex] but they probably plan on proving that at some point.

 

[BrainHurts]

Oh thanks so much! This book doesn't assume topology, but one thing I've always been confused on is that

if

[itex] \overline{D(z_0,r)} = D(z_0,r) \cup \bar{D}(z_0,r) [/itex],
why change the notation? I see you said that they turn out to be equal. Is this to specify a more theoretical idea than a practical idea?

 

[WWGD]

Saying it a bit differently from O.Shredder, it is not immediate that what is called (kind of confusingly) a closed ball--your definition in the bottom --is not a closed set, and, like Office Shredder said, this will be proved at some later point in the book.

 

[blue_raver22]

Hi there,

I can explain the notation for disks in a bit more detail. The notation D(z_0,r) represents an open disk centered at z_0 with radius r. This means that the set includes all points in the complex plane that are within a distance of r from the center point z_0, but does not include the boundary of the disk.

On the other hand, the notation \bar{D}(z_0,r) represents a closed disk, which includes the boundary of the disk as well. This means that the set includes all points in the complex plane that are within a distance of r from the center point z_0, including the points on the boundary.

Now, for the notation \overline{D(z_0,r)}, this represents the closure of an open disk. The closure of a set includes all the points in the set as well as its boundary. So, in this case, it would be the closed disk \bar{D}(z_0,r).

I hope this helps clarify the notation for you. If you have any further questions, please feel free to ask.