Question

Let \( S \) be the set of all rational numbers in \( (0,1) \). Then which of the following statements is/are TRUE?

• A. \( S \) is a closed subset of \( \mathbb{R} \).
• B. \( S \) is not a closed subset of \( \mathbb{R} \).
• C. \( S \) is an open subset of \( \mathbb{R} \).
• D. \( S \) is a limit point of \( S \).

Hint:

A set of rational numbers in an interval is not closed because it does not contain its irrational limit points.

Answer 1

The Correct Option is B, D

Step 1: Analyzing closed subsets.
A set is closed if it contains all its limit points. The set \( S \) is the set of all rational numbers in the open interval \( (0, 1) \). Since rational numbers are dense in the real numbers, any point in \( (0, 1) \), whether rational or irrational, is a limit point of \( S \).
Step 2: Analyzing openness.
The set \( S \) is not open because there are no open intervals around any point in \( S \) that are entirely contained within \( S \), since \( S \) only contains rational numbers and the irrationals are dense in \( (0, 1) \).
Step 3: Conclusion.
Thus, the correct answer is (B) and (D).