A continuous self map on the Cantor space $C = \{0,1\}^\mathbb{N}$ is a mapping $f = (f_i)_{i\in \mathbb{N}}$ such that each $f_i$ is a map from $C$ to $\{0,1\}$ that depends only on a finite number of coordinates.

If for each coordinate there is only a finite number of $f_i$'s that depend on it, it is easily seen that $f$ is also open when restricting its codomain to its range.

If we take the codomain to be the range, what are examples of open continuous maps that are not of this form, if any? Is there a similarly simple way to describe continuous open self maps on Cantor space in general? What about the case where the codomain is not the range but the full Cantor space?

Sorry if this is too elementary for this site.

EDIT: the first question has a trivial answer (simply take a diagonal embedding, see comments). The two other questions remain.