# Dictionary Order Topology on ##\mathbb{R}^2## Metrizable?

## Homework Statement

I am trying to show that there exists a metric on ##\mathbb{R}^2## that induces the dictionary order topology on the plane.

## The Attempt at a Solution

If I recall correctly, vertical intervals in the plane form basis elements for the dictionary order topology. With this in mind, I am trying to define a metric such that the corresponding ##\epsilon##-balls are vertical strips. The only function that I could come up with is

$$d((x,y),(w,z)) = \begin{cases} \infty ,& x \neq w \\ |y-z| ,& x=w \\ \end{cases}$$

Based on my work, it seems that this is a well-defined metric. However, somehow I feel that I am cheating by using ##\infty##. Is this in fact a valid metric?