Consider the following statements: (A) The union of finite numbers of closed set is a closed set (B)…
Finite unions of closed sets and finite intersections of open sets preserve the stated property, so A,D are true. The empty set is both open and closed, making B false.…
Choose your answer
Consider the following statements:
(A) The union of finite numbers of closed set is a closed set
(B) The empty set is an open set but not a closed set
(C) Every infinite set is an open set
(D) The intersection of finite numbers of open sets is an open set
(A) The union of finite numbers of closed set is a closed set
(B) The empty set is an open set but not a closed set
(C) Every infinite set is an open set
(D) The intersection of finite numbers of open sets is an open set
Answer and explanation
Correct answer C
Finite unions of closed sets and finite intersections of open sets preserve the stated property, so A,D are true. The empty set is both open and closed, making B false. Infinite cardinality does not imply openness; the integers inℝ or a closed interval supply counterexamples to C. Hence A,D only. (A) and (B) Only (B) and (C) Only (A) and (D) Only (A), (B) and (C) Only
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UGC NET Economics