Recently in something that I'm studying, I needed to know if the following map is transitive: $\sigma: M^{\mathbb{N}}\to M^{\mathbb{N}}$ the unilateral shift, where $M$ is a uncountable compact metric space. We are considering in $ M^{\mathbb{N}} $ the topology of the cylinders.

1.The case in which $M$ is a finite set is a known fact. The answer is yes and the solution consists in construct a $\underline{x}\in M^{\mathbb{N}}$ just pasting all possible finite sequences of elements of $M$.

2.The second natural case is when $M$ is infinite countable, the solution is similar to the previous case

3.Given the method of item 1, in order to establish the transitivity, the solution of my problem seems to be false, because it seems to me that it will be necessary a non-countable number of iterates to go through all opens of $M^{\mathbb{N}}$

4.Require that $M$ has countable base as hypothesis can help in order to establish transitivity?