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

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.