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Part CCSIR NET December 2024mutual-embedding-does-not-give-homeomorphism

Mutual embedding does not give homeomorphism

Let A, B, C be topological spaces such that A is homeomorphic to B, B is a subspace of C and the closure of B equals C. Let C be homeomorphic to a subspace W of A. Which of the following statements are FALSE?

  1. A.The spaces B, the closure of W, and C are homeomorphic.✓
  2. B.The spaces B, W, C are homeomorphic.✓
  3. C.If C is compact, then A, B, C are homeomorphic.✓
  4. D.If A is connected, then B and C are connected.

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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