 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ib/paper-2/10e/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

The identity map

id : (P_{n}, d_{\infty}) ⟶ (P_{n}, d_{1})

is a continuous bijection by

d_{1} \leq d_{\infty}

. Its domain is compact by part (i), and its codomain is a Hausdorff space because every metric space is Hausdorff. The compact-to-Hausdorff continuous bijection theorem therefore makes the identity a homeomorphism. Consequently

d_{1} and d_{\infty} induce the same topology on P_{n} .

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