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

Every open set contains the special point

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

A.The set of all irrational numbers is dense in $(R, τ)$ .
B.For each prime number p, the set { $0, p$ } is dense in $(R, τ)$ .✓
C.[0, 1] is compact in $(R, τ)$ .
D. $(R, τ)$ 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.

See pricing

50 are analysed free — try those first.

Related counterexample: A compact T₁ space is Hausdorff

Every open set contains the special point

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