Exercise · ZW-TOPOLOGY-OF-THE-COMPLEX-PLANE-007 · ★★☆ Intermediate
Which of these sets is NOT connected?
Choose one option.
Show answer
- A. \(\{z:\operatorname{Re}z\ne0\}\)
Explanation
\(\operatorname{Re}z\neq0\) is two half-planes separated by the imaginary axis; you cannot go from one to the other without leaving the set. The rest are connected.
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-008 Which of these sets is a domain (open and connected)?
- ZW-TOPOLOGY-OF-THE-COMPLEX-PLANE-009 How many points does the intersection of the open disks D(1,1) and D(-1,1) have?
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