1. Show that for k = 2, 3,... , there is a continuous function f (k) from [0, 1] onto the unit cube [0, 1]k in Rk . Hint: Let f (2)(t ) := (g(t ), h(t )) := f (t ) for 0 ≤ t ≤ 1 from Problem 9. For any (x, y, z) ∈ [0, 1]3, there are t and u in [0, 1] with f (u) = (y, z) and f (t ) = (x, u), so f (3)(t ) := (g(t ), g(h(t )), h(h(t ))) = (x, y, z). Iterate this construction.
2. Show that there is a continuous function from [0, 1] onto Tin≥1[0, 1]n , a countable product of copies of [0, 1], with product topology. Hint: Take the sequence f (k) as in Problem 10. Let Fk (t )n := f (k)(t )n for n ≤ k, 0 for n > k. Show that Fk converge to the desired function as k → ∞.