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Part CCSIR NET June 2025continuity-controls-images-of-connected-sets-not-preimages

Continuity controls images of connected sets not preimages

Let p:R→Rp : \mathbb{R} \to \mathbb{R} be a nonconstant polynomial. Which of the following statements are true?

  1. A.The preimage of a compact set under p is a compact set.✓ Correct
  2. B.The preimage of a connected set under p is a connected set.
  3. C.Every point x∈Rx \in \mathbb{R} has an open neighbourhood U_x such that the restriction p∣Uxp|_{U_x} is a homeomorphism onto an open set in R\mathbb{R}.
  4. D.The image of a bounded set under p is a bounded set.✓ 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 converse assumed.

See pricing

50 are analysed free — try those first.

The trap it tests

Converse assumed

The theorem runs one way. You used it in the other.

Drill statements like this

Related counterexample: Bounded ⇒ totally bounded

The chapter behind this: Compactness — the implication map — free to read

From Metric Spaces › Compactness: open covers, sequential, Heine–Borel

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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