Exercise · ZW-TOPOLOGY-OF-THE-COMPLEX-PLANE-010 · ★★★ Advanced
Which statement is true?
Choose one option.
Show answer
- B. \(D(1,1)\subset D(0,2)\)
Explanation
If \(|z-1|<1\), then \(|z|\le|z-1|+1<2\), so \(D(1,1)\subset D(0,2)\). The converse fails: \(-1.5\) is in \(D(0,2)\) but not in \(D(1,1)\).
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)?