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.

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