# The Example of the Uncountable Product of the Unit Interval

Let $I=[0,1]$. In a previous post, I showed that $I^I$ is not hereditarily separable. I would like to make an observation here that both $I^{\omega_1}$ and $I^I$ are also not hereditarily normal and not hereditarily Lindelof. In another previous post, I gave a proof that the uncountable product of the integers, $\mathbb{N}^{\omega_1}$ is not normal. Note that both $I^{\omega_1}$ and $I^I$ contain a copy of $\mathbb{N}^{\omega_1}$. So both $I^{\omega_1}$ and $I^I$ are not completely normal and not hereditarily Lindelof.