Order Isomorphism of Cartesian Products

Question

I've got the following problem to solve:
Are $N\times R$ and $R \times R$ with lexicographical order isomorphic?

Any help much appreciated.

Answer 1

Hint: look at all Dedekind cuts, i.e. all divisions of the set $X$ ($X$ could be $\mathbb N\times\mathbb R$ or $\mathbb R\times\mathbb R$) into two nonempty sets $A, B\subseteq X, A\cup B=X$ such that for every $a\in A, b\in B$ we have $a\lt b$.

For some of those Dedekind cuts $(A,B)$ there will be $x\in X$ such that $x\ge a (a\in A)$ and $x\le b (b\in B)$. For some, there won't be such $x$.

What is the cardinality of the set of Dedekind cuts that don't have the corresponding $x$? Also, is this cardinality an invariant of the order isomorphism?

Answer 2

Connected components in both spaces are of the form {x}×R.
One space has countably many components, the other uncountably many.
If they were order isomorphic they'd be homeomorphic, which they are not.