 Solution

ID: past-exam-of-the-metric-spaces-and-complex-analysis-course-of-the-university-of-oxford/2023/1/a/ii/solution

F \subseteq X

X

when its complement is an open set in the metric topology. Equivalently,

x \in X ∖ F ⟹ \exists r > 0 : B (x, r) \cap F = \emptyset .

The open ball is taken inside the ambient metric space

X

; closedness always depends on that ambient space.

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