Compactness and Connectedness · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Properties that survive continuous maps

Mathematics · Topology · ages 23-24
Name ______________________   Date ____________

  1. A continuous f has f of 0 = -2 and f of 3 = 7. What follows?

    • f must be constant
    • f must be differentiable
    • f has a zero inside 0 to 3

  2. What is the maximum of f(x) = 5 - x squared on [-2, 2]?

    Answer: ______________

  3. A continuous f has f(0) = -2 and f(3) = 7. What follows?

    • f must be constant
    • f attains the value 100
    • f has a zero in (0, 3)
    • f must be differentiable

  4. The continuous image of a compact space is compact.

    Circle one:   True   False

  5. What is the minimum of f(x) = (x - 3) squared + 1 on [0, 5]?

    Answer: ______________

  6. Which of these sets is NOT compact?

    • The open interval (0, 1)
    • The closed interval [0, 1]
    • A three-point set
    • The unit circle

  7. A student says every closed bounded set in R^n is compact. Is that right?

    Circle one:   True   False

  8. A set is closed and bounded but not compact. Where can it live?

    • On the real line
    • In a finite closed interval
    • Outside Euclidean space, as with an infinite dimensional ball

  9. Tom says compactness means closed and bounded in every space. What is wrong?

    • He reversed image and preimage
    • He promoted a Euclidean theorem to every space
    • He confused connectedness with compactness

  10. A student says every finite set is compact. Is that right?

    Circle one:   True   False

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Properties that survive continuous maps W1-mt_ybV1lUedHs-s1

  1. f has a zero inside 0 to 3 · The image is an interval containing -2 and 7, hence 0.
  2. 5 · The peak of 5 - x^2 sits at x = 0 with value 5.
  3. f has a zero in (0, 3) · The intermediate value theorem forces every value between -2 and 7, so f has a zero in (0, 3).
  4. True · Covers pull back and finitely many suffice.
  5. 1 · The vertex at x = 3 gives the minimum value 1.
  6. The open interval (0, 1) · The cover by (1/n, 1) has no finite subcover, so the open interval (0, 1) is not compact.
  7. True · Closed and bounded sets in R^n are compact; that is Heine-Borel.
  8. Outside Euclidean space, as with an infinite dimensional ball · The match of closed-bounded with compact is Euclidean only.
  9. He promoted a Euclidean theorem to every space · The description is Euclidean; the definition is open covers.
  10. True · Every open cover has a finite subcover by pigeonhole over finitely many points.
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