It is well known that there cannot be a homeomorphism between $\mathbb{R}$ and $\mathbb{R^2}$ because if we remove the origin from $\mathbb{R}$ the space becomes disconnected, but if we remove the origin from $\mathbb{R^2}$ the resulting space is connected.
My question is, does there exist an analogous result showing that there cannot be a homeomorphism between $C([0,1])$ and $C([0,1]^2)$ where the sets of continuous functions have the topology induced by the sup-norm $\|f\| = \max_{x} |f(x)|$?