I'm having trouble constructing, and understanding the construction, of a metric that induces the dictionary order topology on $\mathbb R \times \mathbb R$. There are example metrics posted here on Math.SE, but that is not what I'm looking for. I'd like to understand how to approach constructing the proper metric. My first attempt was to define $$d((x_1,x_2),(y_1,y_2)) = \begin{cases} |y_1 – x_1|, & \mbox{if } 0 < |y_1 – x_1| \\ |y_2 – x_2|, & \mbox{if } x_1 = y_1, \mbox{and } 0 < |y_2 – x_2| \\ 0, & \mbox{if } x = y \end{cases}$$ However, even if I didn't make a mistake in my calculations, and this is in fact a metric, I don't know how to approach proving that $d$ induces the dictionary order topology. To summarize, I'd like to know how to approach constructing the metric and if my first attempt even works.
Trouble constructing metric for $\mathbb R \times \mathbb R$ in the dictionary order.
general-topologymetric-spaces
Best Answer
Intuitively, the plane $\Bbb R^2$ in the lexicographic order, is just a set of vertical lines, all of which are topological copies of $\Bbb R$ and such that different vertical lines do not really interact: the horizonal line $\mathbb{R} \times \{0\}$ has the discrete topology as $(x,0)$ has the neighbourhood $O(x):= \{x\} \times (-1,1)$ which equals the open interval $((x,-1),(x,1))$ in the order, such that $O(x) \cap \left(\mathbb{R} \times \{0\}\right) = \{(x,0)\}$. Some more thought will tell you that in fact $\Bbb R^2$ in the order topology is just homeomorphic to the product $(\Bbb R, \mathcal{T}_d) \times (\Bbb R, \mathcal{T}_e)$, where the first factor has the (metrisable) discrete topology and the second the normal Euclidean one. This suggests a metric: we can use the sum (or $\max$-metric) of the component metrics of the factors to get the right product topology, so define
$$d((x_1, y_1), (x_2,y_2)) = \begin{cases} |x_2-y_2| & \text{ if } x_1 = y_1 \\ 1+|x_2-y_2| & \text{ if } x_1 \neq y_1 \\ \end{cases}$$
which is then (by standard facts, I hope for you) a metric that induces the product topology of the discrete metric and the Euclidean metric, and so the right topology for the lexicographically ordered plane.