Topological properties of a countable product of a disconnected subspace of $\mathbb R$

Let \(\omega\) be the cardinality of the natural numbers and let \[ U=[-2,0)\cup[1,2]\subset\mathbb R \] with the subspace topology. For the product space \(U^\omega\), determine which of the following properties hold, giving a reason for each:

1. connected;
2. path-connected;
3. compact;
4. Hausdorff;
5. first countable;
6. metrizable;
7. completely regular;
8. normal.