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Part BCSIR NET June 2025the-only-open-singleton-is-forced-to-be-fixed-leaving-four-points-free

The only open singleton is forced to be fixed leaving four points free

Let S = {1, 2, 3, 4, 5} be equipped with the topology τ=\tau = {∅, {1}, S}. What is the number of homeomorphisms of S onto itself?

  1. A.25
  2. B.120
  3. C.24✓ Correct
  4. D.6

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 hypothesis dropped.

See pricing

50 are analysed free — try those first.

The trap it tests

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this

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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