I was thinking about the correct statement

An open cover $ ([0,1] \cap \Bbb{Q}) \times ([0,1] / \Bbb{Q}) = \{(x,y) \in [0,1] \times [0,1] : x \in \Bbb{Q} , y \notin \Bbb{Q}\} \subset (\Bbb{R}^2, d_{\infty})$ that doesnot have any finite subcover.

Here we see that $ \Bbb{Q} \cap [0,1]$ has no finite subcover but this is is a product of the rationals in $ [0,1]$ and the irrationals.