Question

Match the LIST-I with LIST-II

Choose the correct answer from the options given below:

• A. A-I, B-IV, C-III, D-II
• B. A-III, B-I, C-IV, D-II
• C. A-II, B-III, C-IV, D-I
• D. A-II, B-IV, C-I, D-III

Hint:

The only two subsets of $\mathbb{R}$ that are both open and closed (clopen) are $\emptyset$ and $\mathbb{R}$ itself!

Answer 1

The Correct Option is B

Step 1 : Concept:
This question tests topological classification of subsets of $\mathbb{R}$ under the standard topology (open, closed, clopen, neither).

Step 2 : Key Formulas and Approach:
1. Clopen sets in $\mathbb{R}$ are $\emptyset$ and $\mathbb{R}$. 2. Finite sets contain all their limit points (since they have no limit points), hence closed. 3. Intervals $(a,b)$, $(a,\infty)$, $(-\infty,b)$ are open sets. 4. Dense sets with dense complements in $\mathbb{R}$ (like $\mathbb{Q}$) are neither open nor closed.

Step 3 : Step-by-step Explanation:

• Item A:
The empty set $\emptyset$ is vacuously open and its complement $\mathbb{R}$ is open, so $\emptyset$ is both open and closed (clopen) in $\mathbb{R}$. Matches with III.

• Item B:
Every finite set $S = \{x_1, \dots, x_k\}$ has no limit points ($S' = \emptyset \subseteq S$), so it contains all its limit points and is closed in $\mathbb{R}$. Matches with I.

• Item C:
The set $(0, \infty)$ is an open interval/ray in $\mathbb{R}$, so it is open in $\mathbb{R}$. Matches with IV.

• Item D:
For the set of rational numbers $\mathbb{Q}$: - $\mathbb{Q}$ contains no open intervals, so it is not open. - The closure $\overline{\mathbb{Q}} = \mathbb{R} \neq \mathbb{Q}$, so it is not closed. Thus, $\mathbb{Q}$ is neither open nor closed in $\mathbb{R}$. Matches with II.

Step 4 : Final Answer:
The correct matching is A-III, B-I, C-IV, D-II, which corresponds to option (B).