10. 2^ℕ homeomorphic to 3^ℕ

problem

Show that the spaces $2^ℕ$ and $3^ℕ$ are homeomorphic.

See Brouwer’s theorem in Anush’s DST notes:

theorem

The Cantor space $𝒞=2^ℕ$ is the unique (up to homeomorphism) perfect, nonempty, compact metrizable, zero-dimensional space.

(to make things explicit, read the proof and try to specialize it to $3^ℕ$.)