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Part CCSIR NET December 2023totally-disconnected

Totally disconnected

Consider R2\mathbb{R}^{2} with the Euclidean topology and Q2⊂R2\mathbb{Q}^{2} \subset \mathbb{R}^{2} with the subspace topology. Which of the following statements are true?

  1. A.Q2\mathbb{Q}^{2} is connected.
  2. B.If A is a non-empty connected subset of Q2\mathbb{Q}^{2}, then A has exactly one element.✓ Correct
  3. C.Q2\mathbb{Q}^{2} is Hausdorff.✓ Correct
  4. D.{(x,y)∈Q2(x, y) \in \mathbb{Q}^{2} | x2+y2=1x^{2} + y^{2} = 1} is compact in the subspace topology.

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 property not inherited.

See pricing

50 are analysed free — try those first.

The trap it tests

Property not inherited

A property assumed to pass to subobjects, quotients, or through a chain. It does not.

Drill statements like this

Related counterexample: A product of normal spaces is normal

More on this topic

The chapter behind this: Separation axioms and what survives which construction — free to read

From Topology › Continuity, homeomorphism, separation axioms

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