Correct answer: (A) Every compact subset is closed.; (C) X is T₁ but not Hausdorff.
Explanation
Singletons are closed, proving T₁. Two nonempty opens intersect because a union of countable complements cannot exhaust uncountable X; hence not Hausdorff, proving C. Every infinite subset contains distinct a_n; opens U_k=X∖{a_n:n≥k} cover it with no finite subcover. Thus compact subsets are finite and closed, proving A. X itself is closed but not compact, refuting B. **A,C**.
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