Let $C([0,1]^\omega)$ denote the set of continuous functions $f:[0,1]^\omega \to \mathbb{R}$. We endow $C([0,1]^\omega)$ with two topologies. Let $\tau$ be the pointwise topology on $C([0,1]^\omega)$ and let $\sigma$ be the weak topology.

Are $(C([0,1]^\omega), \tau)$ and $(C([0,1]^\omega), \sigma)$ isomorphic?

Let $C([0,1]^\omega)$ denote the set of continuous functions $f:[0,1]^\omega \to \mathbb{R}$. We endow $C([0,1]^\omega)$ with two topologies. Let $\tau$ be the pointwise topology on $C([0,1]^\omega)$ and let $\sigma$ be the weak topology.

Are $(C([0,1]^\omega), \tau)$ and $(C([0,1]^\omega), \sigma)$ homeomorphic?