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Part CCSIR NET June 2024finer-than-hausdorff-is-still-hausdorff

Finer than hausdorff is still hausdorff

Let τ\tau be the smallest topology on the set R\mathbb{R} containing β=\beta = {[a, b) | a<b;a,b∈Ra < b; a, b \in \mathbb{R}}. Which of the following statements are true?

  1. A.β\beta is a basis for topology τ\tau✓ Correct
  2. B.R\mathbb{R} is compact in the topology τ\tau
  3. C.Topology τ\tau is the same as the Euclidean topology
  4. D.Topology τ\tau is Hausdorff✓ Correct

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 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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