Clarification on an exercise from Munkres about a basis for $\mathbb R^2$

Question

Here is an exercise from Munkres:

Does he mean the product topology on $\mathbb R^2$ with the topology on $\mathbb R$ being the order topology? Or does he mean the dictionary order topology on $\mathbb R^2$? In the former case, the solution is clear because all intervals $(a,b)$ with $a,b$ rational form a basis for the order topology on $\mathbb R$ and the product of bases is a basis for the product topology. In the latter case, I'm not sure how to solve this.

Answer 1

When authors talk about $\mathbb{R}^2$ without qualifications, they mean the Euclidean topology on it, which can be seen as being induced from a metric like $d(x,y) = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2}$, or as the product topology from $\mathbb{R} \times \mathbb{R}$ where $\mathbb{R}$ has its usual (order or metric) topology. It comes down to the same thing.

Answer 2

It's a basis for the product topology on $\mathbb{R}^2$ (which is the same thing as the usual topology on $\mathbb{R}^2$), not for the order topology.