Are $R$ and $\mathbb R×\mathbb R$ similar?

Question

Are $ R$ and $\mathbb R×\mathbb R$ similar ? With antilexicographic order $( (a,b)<(c,d)\iff( b < d)\vee (b=d\text{ and }a <c))$ . Two sets are similar if there exists a bijection between them which keeps the order in sets. I think they aren't but can't find a similarity invariant. They should differ in continuity (every nonempty subset with an upper bound has a supremum ), but how do I prove that for $\mathbb R×\mathbb R$? Edit: Thanks for the hint, I proved it.

Answer 1

Hint The set $\{ (x,0) | x \in \mathbb R \}$ is bounded and in $\mathbb R \times \mathbb R$. Show that it doesn't have a supremum.