 Solution

ID: past-exam-of-the-metric-spaces-and-complex-analysis-course-of-the-university-of-oxford/2014/2/b/solution

a, b

, the elementary inequalities

max (∣ a ∣, ∣ b ∣) \leq a^{2} + b^{2} \leq 2 max (∣ a ∣, ∣ b ∣)

give

δ (p, q) \leq d_{2} (p, q) \leq 2 δ (p, q) .

The identity map between the two versions of the square is bijective, and both it and its inverse are Lipschitz continuous maps by these inequalities. It is therefore a homeomorphism. In fact the same comparison works on all of

R^{2}

; restriction to the square causes no difficulty.

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