{c}

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

Solution

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

 Solution (source code)