Isomorphism of $\mathbb R$ and $\mathbb{R} \times \mathbb R$

Question

How to prove that the set $\mathbb R \times \mathbb R$ with lexicographic order is not isomorphic with $\mathbb R$ with linear order? I know I have to show that there is no such functions $f:\mathbb R \rightarrow\mathbb R \times \mathbb R$ which is an isomorphism but I dont't know how to write this proof.

Answer 1

Two isomorphic ordered sets share all properties that can be expressed in the language of ordered sets.

For instance, if $f\colon X\to Y$ is an order isomorphism and $A$ is a subset of $X$, then $A$ has a least upper bound if and only if $f(A)$ has a least upper bound.

Also, $A$ is upper bounded if and only if $f(A)$ is upper bounded.

Exercise: prove the two statements above.

Now, can you find a subset $A$ of $\mathbb{R}\times\mathbb{R}$ that's upper bounded but has no least upper bound? If $f\colon\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is an isomorphism, can you find a contradiction?