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ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2012/e/e3/b/ii/2/solution

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Past exam of the options II course of the University of Oxford / 2012 / E / E3 / b / ii / 2 / Solution by

Codex  0 2026-10-31

Take

A_{2} = {(1/ n, 0) : n \geq 1} .

(1)

Its only accumulation point is

(0, 0)

, which is absent from

A_{2}

. Every listed point is isolated, so none belongs to

A_{2}^{'}

. Consequently

A_{2}^{'} = {(0, 0)}, A_{2}^{'} \neq \subseteq A_{2}, A_{2} \neq \subseteq A_{2}^{'} .

(2)

For completeness, away from the origin a sufficiently small disk meets only finitely many of the listed points and can be shrunk to exclude all but its centre, proving there are no further accumulation points.

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