25  Subsets of the complex plane

The modulus and real and imaginary parts describe curves and regions in the complex plane.

Visualize the following subsets of \(\mathbb{C}\). Decide which sets are open, closed, or neither.

Exercises

Exercise 25.1 \(\{ z \mid |z-1| = 1 \}\)

Exercise 25.2 \(\{ z \mid |\mathrm{i}z-1| < 1 \}\)

Exercise 25.3 \(\{ z \mid |z-\mathrm{i}| + |z + \mathrm{i}| < 4 \}\)

Exercise 25.4 \(\{ z^2 \mid |z - \mathrm{i}| = 1 \}\)

Exercise 25.5 \(\{ z \mid \Re(e^{\mathrm{i}\pi/2} z) > 0 \}\)

Exercise 25.6 \(\{ z \mid \Im(z^2) > 0 \}\)