SCHWEITZER 2010 Problem 10

Consider the space $\{0,1 \} ^{N}$ with the product topology (where $\{0,1 \}$ is a discrete space). Let $T: \{0,1 \} ^ {\mathbb {N}} \rightarrow \{0,1 \} ^ {\mathbb {N}}$ be the left-shift, ie $(Tx) (n) = x (n+1)$ for every $n \in \mathbb {N}$ . Can a finite number of Borel sets be given: $B_ {1}, \ldots, B_ {m} \subset \{0,1 \} ^ {N}$ such that $$\left \{T ^ {i} \left (B_ {j} \right) \mid i \in \mathbb {N}, 1 \leq j \leq m \right \}$$ the $\sigma$ -algebra generated by the set system coincides with the Borel set system?