Uncountable product of many copies of $\mathbb{Z}$ is not paracompact

Question

Let $(X,\tau)$ be the product of uncountably many copies of $\mathbb{Z}$. Prove that $(X,\tau)$ is not paracompact.

In order to prove that something is not paracompact. We need to find an open cover that does not have an open locally finite refinement. How would this open cover look like in uncountable product of $\mathbb{Z}$?

Answer 1

To find facts about spaces, you can try looking it up at

Pi Base

In this case, it says your space is T_2 and not normal, but paracompact+T_2 implies normal, so it is not paracompact.