Prove that ordered sets $\langle \mathbb{R} \times \mathbb{R}, \le_\text{lex} \rangle$ and $\langle \mathbb{R} \times \mathbb{Q}, \le_\text{lex} \rangle$ are not isomorphic ($\le_\text{lex}$ means lexicographical order).
I know that to prove that ordered sets are isomorphic, I would make a monotonic bijection, but how to prove they aren't isomorphic?
Any open interval of $\mathcal{A}:=\langle \mathbb{R} \times \mathbb{R}, \le_\text{lex} \rangle$ contains uncountably many elements, while some open intervals of $\mathcal{B}:=\langle \mathbb{R} \times \mathbb{Q}, \le_\text{lex} \rangle$ have only countably many elements. Hence they're not isomorphic.
Alternative way: give $\mathcal{A},\mathcal{B}$ order topology. Then some closed interval of $\mathcal{A}$ is compact, while no nontrival closed interval of $\mathcal{B}$ is compact. Since any order-preserving bijection is a homeomorphism, $\mathcal{A}$ and $\mathcal{B}$ don't have the same order type.