Part CCSIR NET December 2023totally-disconnected

Totally disconnected

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

A. $Q^{2}$ is connected.
B.If A is a non-empty connected subset of $Q^{2}$ , then A has exactly one element.✓
C. $Q^{2}$ is Hausdorff.✓
D.{ $(x, y) \in Q^{2}$ | $x^{2} + y^{2} = 1$ } 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

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

From Topology › Continuity, homeomorphism, separation axioms

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