Correct answer: (B) Each A_(m,n) is closed; (C) Hausdorff; (D) Metrizable
Explanation
Each residue class is open and its complement is a finite union of the other residue classes modulo |n|, so it is closed. Even and odd classes disconnect the space. Distinct integers can be separated using different classes modulo n>|x−y|, proving Hausdorffness. An explicit compatible metric is $d(x,y)=\sum_{n\ge1}2^{-n}\mathbf1_{x\not\equiv y\pmod n}$. Its finite-coordinate neighborhoods are intersections of residue classes and hence classes modulo their least common multiple. This proves metrizability. **B,C,D**.
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