Is there a cardinal $\kappa$ for which $\mathbb{R}^\kappa$ is not normal?

Question

I'd like to know if normality is a multiplicative property for $\mathbb{R}$ or there is a cardinal $\kappa$ for which $\mathbb{R}^\kappa$ is not normal.

Answer 1

It is well known that $\mathbb{N}^{\aleph_1}$ is not normal (see this answer), and this also holds for higher powers (as lower powers embed as closed subset in the higher ones, and normality is closed hereditary). This implies that any uncountable product of copies of $\mathbb{R}$ is not normal as well (as the powers of $\mathbb{N}$ from above are closed subsets again).