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Let $A$ and $B$ be metric spaces, and let $f: A \rightarrow B$. Suppose that whenever $X$ is an open set in $B$, the set $\{a \in A: f(a) \notin X\}$ is closed in $A .$ Which of the following must be true?

I. $f$ is injective.
II. $f$ is continuous.
III. $f$ is a homeomorphism.

(A) None
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III
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