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Part BCSIR NET December 2024coupled-indices-escape-independent-ones-accumulate

Coupled indices escape independent ones accumulate

Let N\mathbb{N} be the set of positive integers. Consider R2\mathbb{R}^{2} with the Euclidean topology and the subsets A = {(n,1/n):n∈N(n, 1/n) : n \in \mathbb{N}} and B = {(n,1/m):n,m∈N(n, 1/m) : n, m \in \mathbb{N}}. Which of the following statements is true?

  1. A.A is a closed subset of R2\mathbb{R}^{2} but B is not a closed subset of R2\mathbb{R}^{2}.✓ Correct
  2. B.B is a closed subset of R2\mathbb{R}^{2} but A is not a closed subset of R2\mathbb{R}^{2}.
  3. C.Both A and B are closed subsets of R2\mathbb{R}^{2}.
  4. D.Neither A nor B is a closed subset of R2\mathbb{R}^{2}.

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests dependence misread.

See pricing

50 are analysed free — try those first.

The trap it tests

Dependence misread

Independence, pairwise vs mutual, or correlation vs dependence.

Drill statements like this

Related counterexample: An arbitrary intersection of open sets is open

More on this topic

The chapter behind this: Open and closed sets, closure, interior, boundary — free to read

From Metric Spaces › Open/closed sets, limit points, closure, interior

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