Help sketching 'Jungle River Metric' in $\mathbb{R}^2$

Solution 1:

I got same as you for the set $A$. But for set $B$ it's only the vertical line segment connecting $(2,0)$ to $(2,2)$ [including endpoints]. Using $(x_1,x_2)$ for the coordinates of $x$ and $(y_1,y_2)=(2,1)$, the condition $x_1 \neq y_1$ becomes $x_1 \neq 2$, and the distance being at most $1$ becomes the inequality $$|x_2|+1+|x_1-2| \le 1,$$ which is only satisfied at $(x_1,x_2)=(2,0).$ This is already in conflict with the fact that we're at this point doing the case $x_1 \neq 2,$ but even throwing it in has no effect as it's already on the segment joining $(2,0)$ to $(2,2).$