Show that R^2 with the dictionary order topology is homeomorphic to Rd * R.

Question

I am currently working topology and I can not prove this exercise.

Let $Rd$ denote the set $R$ with the discrete topology. Show that $R^2$ with the dictionary order topology is homeomorphic to $Rd \times R$.

Answer 1

The base set {r}×(a,b) of Rd×R is the open set
((r,a), (r,b)) of the dictionary order of R×R.
The homeomorphism is the identity map.