Separating class for $C[0,1]$

Question

The finite dimensional sets $\mathcal{C}:=\{\pi_{t_1,\ldots, t_d}^{-1}(B) : {t_1,\ldots, t_d} \in [0,1]$ and $B \in \mathcal{B}(\mathbb{R}^d)\}$ forms a separating class for $C[0,1]$. This is proved by showing that $\mathcal{C}$ forms a $\pi$-system and that they generate the Borel $\sigma-$-algebra $\mathcal{B}_{C[0,1]}$. I was wondering if the system $\mathcal{C}_0:=\{\pi_{t}^{-1}(B) : t\in [0,1]$ and $B \in \mathcal{B}(\mathbb{R})\}$ is also a determining class since it also generate $\sigma-$-algebra $\mathcal{B}_{C[0,1]}$. Since I cannot find it i'd excpect that this is not true. Isn't $\mathcal{C}_0$ a $\pi$-system either?

Answer 1

$\mathcal C_0$ is not a $\pi$ system. For example $\{f\in C[0,1]: f(0)>0\} \cap \{f\in C[0,1]: f(1)>0\}$ cannot be written as $\{f\in C[0,1]: f(t) \in B \}$ for any Borel set $B$. ( You can always find a continuous function f such that $f(0)=f(1)=1$ but $f(t) \notin B$ since $B$ cannot be $\mathbb R$).