Open Sets, Closed Sets and Limit Points · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Open sets, closed sets, and continuity

Mathematics · Topology · ages 20-21
Name ______________________   Date ____________

  1. The closure of the open interval (0, 1) adds the endpoints. How many points are added?

    Answer: ______________

  2. In the reals with the usual topology, what is the interior of the closed interval [0, 1]?

    • [0, 1]
    • (0, 1)
    • The two point set of endpoints
    • The empty set

  3. In the reals with the usual topology, what is the interior of the closed interval from 0 to 1?

    • The same closed interval
    • The open interval without endpoints
    • Only the two endpoints

  4. Arbitrary unions of open sets are open, but only finite intersections are guaranteed open.

    Circle one:   True   False

  5. Consider the sequence points one over n for positive integers n. The closure adds exactly the limit point. How many points does the closure add?

    Answer: ______________

  6. Which condition matches continuity of f?

    • Images of open sets are always open
    • Only constant maps are continuous
    • Preimages of open sets are open

  7. Why can infinite intersections of open sets fail to be open?

    • Nested intervals (minus 1/n, 1/n) intersect to the singleton zero, which is not open
    • Infinite unions of opens always fail first
    • Intersections of opens are always open
    • Open sets are always bounded

  8. Every finite intersection of open sets is open. True or false?

    Circle one:   True   False

  9. f is 0 at and left of 0, and 1 right of 0. Which open preimage witnesses the jump at 0?

    • Every preimage stays open here
    • No open set detects the jump
    • Values near 0 pull back to a half line that is not open

  10. Let f be zero at and left of zero and one right of zero. Which open preimage witnesses the discontinuity at zero?

    • The preimage of (0.5, 1.5) is (0, infinity), which is not open
    • Every preimage of an open set is open here
    • The preimage of (minus 0.5, 0.5) is (minus infinity, 0], which is not open
    • The preimage of the singleton zero is empty
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Answer key

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

Open sets, closed sets, and continuity W1-mt_9bBdT38dbB-s1

  1. 2 · Zero and one join, so two points are added.
  2. (0, 1) · Endpoints have no ball inside the interval, so only (0, 1) survives.
  3. The open interval without endpoints · Endpoints have no ball inside the interval, so only the middle survives.
  4. True · Infinite intersections can collapse to a non open set.
  5. 1 · Only the limit joins the sequence.
  6. Preimages of open sets are open · Open preimages characterise continuity; the image versions fail in general.
  7. Nested intervals (minus 1/n, 1/n) intersect to the singleton zero, which is not open · The shrinking intervals trap only zero, and no ball around zero fits inside a point.
  8. True · True. Take the smallest of the finitely many witness balls.
  9. Values near 0 pull back to a half line that is not open · That half line holds 0 with no room right, so it is not open.
  10. The preimage of (minus 0.5, 0.5) is (minus infinity, 0], which is not open · The half line (minus infinity, 0] contains 0 with no room to the right, so it is not open.
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