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Part CCSIR NET December 2024hausdorff-pulls-back-it-does-not-push-forward

Hausdorff pulls back it does not push forward

Let X and Y be topological spaces and f : X → Y be a continuous function. Which of the following statements are true?

  1. A.If X and Y are compact, Y is Hausdorff and f is onto, then X is also Hausdorff.
  2. B.If X is an infinite compact set and f is a homeomorphism from X to f(X) (where f(X) is given the subspace topology), then Y is compact.
  3. C.If X is Hausdorff and f is onto, then Y is Hausdorff.
  4. D.If f is a homeomorphism, then X is second-countable if and only if Y is second-countable.✓

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.

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50 are analysed free — try those first.

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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