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Part CCSIR NET December 2024every-open-set-contains-the-special-point

Every open set contains the special point

Define a topology τ\tau on R\mathbb{R} as follows: a subset U of R\mathbb{R} is in the topology τ\tau if and only if U = ∅ or 0 ∈ U. Which of the following statements are true?

  1. A.The set of all irrational numbers is dense in (R,τ)(\mathbb{R}, \tau).
  2. B.For each prime number p, the set {0,p0, \sqrt{p}} is dense in (R,τ)(\mathbb{R}, \tau).✓ Correct
  3. C.[0, 1] is compact in (R,τ)(\mathbb{R}, \tau).
  4. D.(R,τ)(\mathbb{R}, \tau) is Hausdorff.

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 standard counterexample.

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The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Related counterexample: A compact T₁ space is Hausdorff

More on this topic

The chapter behind this: The counterexample zoo: cofinite, cocountable, Sorgenfrey, Cantor — free to read

From Topology › Standard spaces: cofinite, cocountable, Sorgenfrey, Cantor set

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