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

For closed sets in $\mathbb{R}$, the answers are $X_1$: no; $X_2$: yes; $X_3$: yes. The first interval omits its limit point zero. The second contains its finite limit points, and the empty set is closed in every topology.

For open sets in $\mathbb{R}$, the answers are $X_1$: yes; $X_2$: no; $X_3$: yes. Every point of the positive ray has a small interval contained in it, while zero is not an interior point of the nonnegative ray. The empty set is open by the topology axioms.

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.