Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2009 / 4 / i / c

For open sets in $\mathbb{C}$, the answers are $X_1$: no; $X_2$: no; $X_3$: yes. Every complex disk centered at a real point contains nonreal points, so a nonempty subset of the real axis cannot be open in the complex plane. The empty set remains open.