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

Compact domains tame functions

Mathematics · Topology · ages 21-22
Name ______________________   Date ____________

  1. Function negates the square then adds six times the input. On the closed interval from 0 to 6, what is its maximum value?

    Answer: ______________

  2. Which interval guarantees every continuous real function is bounded and attains its bounds?

    • (0, 1)
    • [0, 1]
    • Rationals in [0, 1]
    • (0, 1]

  3. Which set guarantees every continuous real function is bounded and attains its bounds?

    • (0, 1)
    • [0, 1]
    • Rationals in [0, 1]

  4. The continuous image of a compact space is compact, hence bounded in the reals.

    Circle one:   True   False

  5. Why can maxima fail on the half open interval 0 excluded to 1?

    • They may escape toward the missing endpoint
    • They always converge inside anyway
    • They become eventually constant

  6. Heine Borel says closed bounded intervals in real numbers are compact, and completeness of reals is used to get limit points. Is this roughly correct?

    Circle one:   True   False

  7. Which set shows closed and bounded does not imply compact in infinite dimensions?

    • Closed interval [0, 1] in real numbers
    • Finite set with 5 points
    • Single point
    • Closed unit ball in infinite dimensional normed space

  8. Which set shows closed plus bounded does not imply compact beyond finite dimensions?

    • Closed interval [0, 1] in the reals
    • A finite set with 5 points
    • Closed unit ball in infinite dimensions

  9. Which bounded set fails compactness by missing limits?

    • Rationals in [0, 1], missing irrational limits
    • (0, 1), missing both endpoints
    • [0, 1], which is compact

  10. Closed interval from 0 to 10 is covered by closed intervals of length 2. How many such intervals suffice to cover it?

    Answer: ______________

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Answer key

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Compact domains tame functions W1-mt_dBLXU0wSN9-s1

  1. 9 · Vertex at 3 gives minus 9 plus 18, equal to 9, which is the maximum.
  2. [0, 1] · [0, 1] is compact, so the extreme value theorem applies, while each other listed set allows unbounded or nonattained examples.
  3. [0, 1] · The closed interval is compact, so maxima are caught inside.
  4. True · Compactness travels along continuous maps, and compact reals are bounded.
  5. They may escape toward the missing endpoint · Escape to the gap breaks the extreme value promise.
  6. True · Closed and bounded gives sequential compactness via Bolzano Weierstrass, which needs completeness.
  7. Closed unit ball in infinite dimensional normed space · Closed unit ball in infinite dimensional normed space is closed and bounded but not compact, while the interval, the finite set, and the singleton are compact.
  8. Closed unit ball in infinite dimensions · That ball is closed and bounded yet holds a sequence with no convergent subsequence.
  9. Rationals in [0, 1], missing irrational limits · Missing limits breaks sequential compactness even inside a bounded fence.
  10. 5 · Length 10 divided by length 2 gives 5 intervals.
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