{c}

For <open sets> in $\mathbb C$, the answers are \b[$X_1$: 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.

Solution

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

 Solution (source code)