1. Show that the closure of a nowhere dense set is nowhere dense.
2. Let (S, d) and (V, e) be two metric spaces. On the Cartesian product S × V take the metric ρ((x, u), (y, v)) = d(x, y) + e(u, v).
Show that the completion of S × V is isometric to the product of the completions of S and of V .
3. Show that the intersection of the complement of a set of first category with a non-empty open set in a complete metric space is not only non-empty but uncountable. Hint: Are singleton sets {x } nowhere dense?