Exercise · ZW-TOPOLOGY-OF-THE-COMPLEX-PLANE-009 · ★★★ Advanced
How many points does the intersection of the open disks \(D(1,1)\) and \(D(-1,1)\) have?
Choose one option.
Show answer
- A. \(0\)
Explanation
If \(z\) were in both, \(2=|(z+1)-(z-1)|\le|z+1|+|z-1|<2\), absurd. The disks only touch at \(0\), which belongs to neither.
Review the concept: Topology of the complex plane
The structure that open disks \(D\) of center \(z_0\) and radius \(r\) give to \(\mathbb{C}\); with them one defines when a set of points \(z\) is open, closed, bounded or connected.
\[D(z_0,r)=\{z\in\mathbb{C}:|z-z_0|
More practice problems: topology of the complex plane
- ZW-TOPOLOGY-OF-THE-COMPLEX-PLANE-001 Which of these points lies in the open disk D(1+i,2)?
- ZW-TOPOLOGY-OF-THE-COMPLEX-PLANE-002 The set \z:|z+2i|<3\ is an open disk. What are its center and radius?
- ZW-TOPOLOGY-OF-THE-COMPLEX-PLANE-003 Which of these sets is open?
- ZW-TOPOLOGY-OF-THE-COMPLEX-PLANE-004 Which of these sets is closed?
- ZW-TOPOLOGY-OF-THE-COMPLEX-PLANE-005 What is the boundary of the disk \z:|z|<1\?
- ZW-TOPOLOGY-OF-THE-COMPLEX-PLANE-006 Which of these sets is bounded?
- ZW-TOPOLOGY-OF-THE-COMPLEX-PLANE-007 Which of these sets is NOT connected?
- ZW-TOPOLOGY-OF-THE-COMPLEX-PLANE-008 Which of these sets is a domain (open and connected)?
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