Complex Made Simple: Notation on Disks

[BrainHurts]

Hello,

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

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

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

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 \overline{D(z_0,r)} mean? Any thoughts?

 

[Office_Shredder]

That means the topological closure of the set D(z_0, r). It turns out to be equal to \overline{D}(z_0, r) 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

\overline{D(z_0,r)} = D(z_0,r) \cup \bar{D}(z_0,r),
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.