Homeomorphism between $(0,1)^{\mathbb{R}}$ and $\mathbb{R}^{\mathbb{R}}$

Question

We already know that $(0,1)$ is homeomorphic to $\mathbb{R}$.Is it possible to generalize this homeomorphism and say that also $(0,1)^{\mathbb{R}}$ is homeomorphic to $\mathbb{R}^{\mathbb{R}}$ if the answer is yes , why we can do that ??

Answer 1

Hint: Prove that for homeomorphic topological spaces $X,Y$ their product topologies $X^I, Y^I$ (for any $I$) are homeomorphic as well.

(Extend the homeomorphism $\pi \colon X \to Y$ coordinate wise to a homeomorhpism $\pi^I \colon X^I \to Y^I$.)