$\mathbb R^{\omega}$ in the box topology not paracompact?

Question

I know that we can't prove that $\mathbb R^{\omega}$ in the box topology is normal, so we can't say for sure that it is paracompact(it is Hausdorff, and every paracompact Hausdorff space is normal). Then can we prove that it is 'not' paracompact?

Answer 1

In The box product of countably many compact metric spaces, Gen. Top. Appl. $2$ ($1971$), pp. $293$-$298$, Mary Ellen Rudin showed that the continuum hypothesis implies that the box product of countably many $\sigma$-compact, locally compact metric spaces is paracompact. In particular, then it implies that the box product of countably many copies of $\Bbb R$ is paracompact, so we cannot prove in $\mathsf{ZFC}$ that this box product is not paracompact.