Question

Which one of the following statements is incorrect?

• A. The set \(\left\{\dfrac{1}{n} : n \in \mathbb{N}\right\}\) is neither an open set nor a closed set in \(\mathbb{R}\).
• B. The set \(\left\{x \in \mathbb{R} : \sin\left(\dfrac{1}{x}\right) = 0\right\}\) is a closed set in \(\mathbb{R}\).
• C. The set \(\{(x,y) \in \mathbb{R}^2 : xy = 0\}\) has no interior points.
• D. The limit points of the set \(\left\{\dfrac{1}{n} + \dfrac{1}{m} : n,m \in \mathbb{N}\right\}\) is \(\{0\} \cup \left\{\dfrac{1}{k} : k \in \mathbb{N}\right\}\).

Hint:

Check whether 0, the limit of the sin(1/x)=0 solutions, actually belongs to that set.

Answer 1

The Correct Option is B

Step 1: Test option (A). The set \(\{1/n : n\in\mathbb{N}\}=\{1,1/2,1/3,\dots\}\) consists of isolated points, so no point has a whole neighborhood contained in the set, hence it is not open. Also \(0\) is a limit of the sequence \(1/n\) but \(0\) is not in the set, so it does not contain all its limit points, hence not closed either. So (A) is true.

Step 2: Test option (B). The set is \(T=\{x\in\mathbb{R}\setminus\{0\} : \sin(1/x)=0\}=\{1/(k\pi) : k \in \mathbb{Z}, k\ne0\}\). As \(k \to \infty\), \(1/(k\pi)\to 0\), so \(0\) is a limit point of \(T\). But \(x=0\) is excluded from \(T\) since \(\sin(1/x)\) is undefined there, so \(T\) does not contain this limit point. Hence \(T\) is NOT closed, so statement (B) is false.

Step 3: Test option (C). The set \(\{(x,y): xy=0\}\) is the union of the two coordinate axes. Any open disc centered at a point of this union contains points with \(xy \ne 0\), so no point is interior. Hence (C) is true.

Step 4: Test option (D). Fix \(m\) and let \(n \to \infty\): the terms \(1/n+1/m \to 1/m\), so every \(1/m\) is a limit point. Letting both \(n,m\to\infty\) gives terms tending to \(0\), so \(0\) is also a limit point, and no other limit points arise. So the limit points are exactly \(\{0\}\cup\{1/k : k\in\mathbb{N}\}\), and (D) is true.

Step 5: Conclusion. Only statement (B) is false.

\[\boxed{\text{Option (B) is incorrect}}\]