 Solution

ID: past-exam-of-the-metric-spaces-and-complex-analysis-course-of-the-university-of-oxford/2014/1/a/ii/solution

\overline{A}

is the smallest closed subset of

M

containing

A

, equivalently the intersection of all such closed subsets. In a metric space it is characterized by

x \in \overline{A} ⟺ B (x, r) \cap A \neq = \emptyset for every r > 0.

If some ball misses

A

, its centre lies outside a closed superset of

A

; conversely the complement of any closed superset is an open neighbourhood disjoint from

A

. This verifies the equivalence of the two descriptions.

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