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Let X be a subset of R^n. Prove that the following are equivalent: 1) X is open...

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Advance Math

Let X be a subset of R^n. Prove that the following areequivalent:

1) X is open in R^n with the Euclidean metricd(x,y) = sqrt((x1 - y1)^2+(x2 - y2)^2+...+(xn - yn)^2)

2) X is open in R^n with the taxicab metricd(x,y)= |x1 - y1|+|x2 - y2|+...+|xn - yn|

3) X is open in R^n with the square metricd(x,y)= max{|x1 - y1|,|x2 - y2|,...,|xn -y n|}

(This can be proved by showing the 1 implies 2 implies 3)

(TOPOLOGY)

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