Let X be a subset of R^n. Prove that the following are
equivalent:
1) X is open in R^n with the Euclidean metric
d(x,y) = sqrt((x1 - y1)^2+(x2 - y2)^2+...+(xn - yn)^2)
2) X is open in R^n with the taxicab metric
d(x,y)= |x1 - y1|+|x2 - y2|+...+|xn - yn|
3) X is open in R^n with the square metric
d(x,y)= max{|x1 - y1|,|x2 - y2|,...,|xn -y n|}
(This can be proved by showing the 1 implies 2 implies 3)...