$f: \prod_{i \geq 1} \{0,1,2,3\} \rightarrow \prod_{i \geq 1} \{0,2\}$ is a homeomorphism

Question

We view $X = \{0,1,2,3\}$ as a topological space equipped with the discrete topology.

$f: \prod_{i \geq 1} X \rightarrow \prod_{i \geq 1} \{0,2\}$ is a homeomorphism.

So f maps a sequence $(a_{i})_{i \geq 1}$ to another sequence $(b_{j})_{j \geq 1}$ with each $a_{i} \in \{0,1,2,3\}$ and each $b_{j} \in \{0,2\}$.

But I don't know how to construct such a homeomorphism f. If I send all the $a_{i}$ to $0$ if j is odd and to $2$ if j is even, then it's not a bijection I think.

Answer 1

Hint. $4=2\times2$. $\qquad$ :-)

Answer 2

Both are homeomorphic to the countable Cantor cube, as they are compact metric zero-dimensional and have no isolated points.