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Part CCSIR NET June 2024cofinite-is-always-compact-never-hausdorff

Cofinite is always compact never hausdorff

Let X denote the topological space R\mathbb{R} with the cofinite topology (i.e., the finite complement topology) and let Y denote the topological space R\mathbb{R} with the Euclidean topology. Which of the following statements are true?

  1. A.X × [0, 1] is closed in X × Y with respect to the product topology✓ Correct
  2. B.X × [0, 1] is compact with respect to the product topology✓ Correct
  3. C.X is compact✓ Correct
  4. D.X × Y is compact with respect to the product 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 standard counterexample.

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50 are analysed free — try those first.

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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