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. In R with its usual metric, how many limit points does A={-4,1,7} have?

    Answer: ______________

  2. Let X={a,b,c} have the discrete metric: distinct points are distance 1 apart. Which statement about {a} is true?

    • It is open in X but is not closed in X.
    • It is both open and closed in X.
    • It is closed in X but is not open in X.
    • It is neither open nor closed in X.

  3. In R with its usual metric, what is the interior of [0,1]?

    • The closed interval [0,1].
    • The open interval (0,1).
    • The set containing only 0 and 1.

  4. In every metric space, arbitrary unions and finite intersections of open sets are open. An infinite intersection need not be open.

    Circle one:   True   False

  5. In R with its usual metric, let A=(-2,0) union (0,3). How many points lie in the interior of the closure of A but not in A?

    Answer: ______________

  6. For a map f:X->Y between metric spaces, which condition is equivalent to continuity?

    • The image of every open set in X is open in Y.
    • The preimage of every singleton in Y contains exactly one point of X.
    • The preimage of every open set in Y is open in X.

  7. A proof takes x in a union of open sets and tries to choose a radius that works in every member. Which change gives a valid proof?

    • Use a ball from one member containing x; it also lies in the union.
    • Take the minimum radius over all members, including those not containing x.
    • Require x to belong to every member before choosing a ball in the union.
    • Use the largest member radius, even if its ball leaves the entire union.

  8. In any metric space, the intersection of an arbitrary family of closed sets is closed.

    Circle one:   True   False

  9. Let f:R->R be 0 for x<=0 and 1 for x>0. For V=(-0.5,0.5), which statement about f^-1(V) witnesses discontinuity?

    • It is (0,infinity); no positive input has a ball contained in this set.
    • It is all of R; both constant pieces make f continuous at 0.
    • It is (-infinity,0]; no ball about 0 lies inside this set.

  10. For f:X->Y between metric spaces, assume every open target set has an open preimage. Fix a in X and epsilon>0. Which construction proves continuity at a?

    • Take a target ball B_Y(f(a),epsilon); use epsilon itself as the source radius without examining any preimage.
    • Take f(B_X(a,epsilon)); call this image open and use a target ball inside it to choose the source radius.
    • Take G=f^-1(B_Y(f(a),epsilon)); openness at a gives delta>0 with B_X(a,delta) inside G.
    • Take G=f^-1({f(a)}); claim that this preimage must be open and contains a source ball for every epsilon.
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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. 0 · The finite set has no limit points. Its members can be isolated, and every outside point has a ball missing the set.
  2. It is both open and closed in X. · B_X(a,1/2)={a}, so {a} is open. Its complement {b,c} is also open, so {a} is closed.
  3. The open interval (0,1). · Points strictly between 0 and 1 have balls inside [0,1]. Neither endpoint does, so the interior is (0,1).
  4. True · A union uses a ball from one member. A finite intersection uses a positive minimum radius. Infinite intersections have no such guarantee.
  5. 1 · The closure is [-2,3], with interior (-2,3). Only 0 belongs to that interior but not to A.
  6. The preimage of every open set in Y is open in X. · Continuity gives open preimages. Conversely, openness of target-ball preimages supplies the source balls required for continuity.
  7. Use a ball from one member containing x; it also lies in the union. · A union point belongs to at least one member. That open member supplies a ball contained in the union.
  8. True · The complement is a union of open complements, hence open. Therefore the intersection is closed.
  9. It is (-infinity,0]; no ball about 0 lies inside this set. · V contains 0 but excludes 1. Its preimage is (-infinity,0], which is not open at 0. Thus f is not continuous.
  10. Take G=f^-1(B_Y(f(a),epsilon)); openness at a gives delta>0 with B_X(a,delta) inside G. · The open target ball has an open preimage containing a. A source delta-ball inside that preimage gives the continuity implication.
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