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Quiz: Point set topology

Page history last edited by RH 11 years, 11 months ago

TThis quiz is designed to test your knowledge of point set topology notions in metric spaces, such as open and closed sets, compact and connected sets, interior and adherent points, etc.

 

Discuss this quiz 

(Key: correct, incorrect, partially correct.)

 

  1. Let Formula be a subset of a metric space Formula. What does it mean for Formula to be an adherent point of Formula?
    1. For every Formula, there exists a Formula such that Formula.
      • CORRECT.
    2. For every Formula, there exists a Formula such that Formula.
      • INCORRECT. This is what it means for Formula to be a limit point of Formula, not an adherent point.
    3. There exists Formula and Formula such that Formula.
    4. For every Formula and all Formula, we have Formula.
    5. For every Formula, there exists Formula such that Formula.
    6. There exists Formula such that Formula for all Formula.
    7. There exists Formula such that Formula for all Formula.
  2. Let E be a subset of a metric space Formula. What does it mean for Formula to not be an adherent point of Formula?
    1. There exists an Formula such that Formula for all Formula.
      • CORRECT.
    2. There exists an Formula and Formula such that Formula.
    3. For every Formula there exists Formula such that Formula.
    4. For every Formula we have Formula for every Formula.
    5. For every Formula there exists an Formula such that Formula.
    6. There exists Formula such that Formula for every Formula.
    7. There exists an Formula such that Formula whenever Formula.
  3. Let Formula be a subset of a metric space Formula. What does it mean for Formula to be an adherent point of Formula?
    1. There exists a sequence Formula in Formula which converges to Formula.
      • CORRECT.
    2. There exists a sequence Formula in Formula which converges to Formula.
      • INCORRECT. This is what it means for x to be a limit point of Formula, not an adherent point.
    3. Every sequence Formula in Formula which converges, converges to Formula.
    4. Every sequence Formula in Formula converges to Formula.
    5. Every Cauchy sequence Formula in Formula converges to Formula.
    6. There exists a sequence Formula in Formula which is a Cauchy sequence.
    7. Every sequence Formula which converges to Formula, must lie in Formula.
  4. Let Formula be a subset of a metric space Formula, let Formula be a function from Formula to another metric space Formula, let Formula be an adherent point of Formula, and let Formula be a point in Formula. What does it mean for Formula to equal Formula?
    1. For every Formula, there exists a Formula such that Formula for all Formula for which Formula.
    2. For every Formula, there exists a Formula such that Formula for all Formula for which Formula.
    3. For every Formula, there exists a Formula such that Formula for all Formula for which Formula.
      • CORRECT.
    4. For every Formula and Formula, we have Formula for all Formula for which Formula.
    5. For every Formula and every Formula, there exists Formula such that Formula if Formula.
    6. For every Formula, there exists a Formula such that Formula and Formula for some Formula.
    7. For every Formula, there exists a Formula such that Formula and Formula for all Formula.
  5. Let Formula be a subset of Formula, let Formula be a function, and let Formula be an element of Formula. What does it mean for Formula to be continuous at Formula?
    1. For every Formula, there exists a Formula such that Formula for all Formula for which Formula.
    2. For every Formula, there exists a Formula such that Formula for all Formula for which Formula.
    3. For every Formula, there exists a Formula such that Formula for all Formula for which Formula.
      • CORRECT.
    4. For every Formula and Formula, we have Formula for all Formula for which Formula.
    5. For every Formula and every Formula, there exists Formula such that Formula if Formula.
    6. For every Formula, there exists a Formula such that Formula and Formula for some Formula.
    7. For every Formula, there exists a Formula such that Formula and Formula for all Formula.
  6. Let Formula be a subset of a metric space Formula, let Formula be a function from Formula to another metric space Formula, and let Formula be an adherent point of Formula. What does it mean for Formula to equal Formula?
    1. For every sequence Formula in Formula which converges to Formula, the sequence Formula converges to Formula.
      • CORRECT.
    2. There exists a sequence Formula in Formula converging to Formula, such that Formula converges to Formula.
    3. Whenever Formula converges to Formula, the sequence Formula , must then converge to Formula.
    4. For every sequence Formula in Formula which converges to Formula, the sequence Formula converges to Formula.
    5. Whenever Formula converges to Formula, the sequence Formula, must lie in Formula and converge to Formula.
    6. Whenever Formula converges to Formula, the sequence Formula, must lie in Formula and converge to Formula.
    7. For every sequence Formula in Formula which converges to Formula, the sequence Formula converges to Formula.
  7. Let Formula be a subset of a metric space Formula, let Formula be a function from Formula to another metric space Formula, and let Formula be an element of Formula. What does it mean for Formula to be continuous at Formula?
    1. For every sequence Formula in Formula which converges to Formula, the sequence Formula converges to f(x).
      • CORRECT.
    2. There exists a sequence Formula in Formula converging to Formula, such that Formula converges to Formula.
    3. Whenever Formula converges to Formula, the sequence Formula, must then converge to Formula.
    4. Every sequence Formula which is convergent, converges to Formula.
    5. Whenever Formula converges to Formula, the sequence Formula, must lie in Formula and converge to Formula.
    6. Whenever Formula converges to Formula, the sequence Formula, must lie in Formula and converge to Formula.
    7. For every sequence Formula in Formula, the sequence Formula converges to Formula.
  8. Let Formula be a function from a metric space Formula to a metric space Formula. What does it mean for Formula to be continuous on Formula?
    1. For every Formula and Formula, there exists a Formula such that Formula for all Formula for which Formula.
      • CORRECT.
    2. For every Formula, there exists a Formula such that Formula for all Formula for which Formula.
      • INCORRECT. This is what it means for Formula to be uniformly continuous on Formula.
    3. For every Formula, there exists a Formula and Formula such that Formula and Formula.
    4. For every Formula and Formula, there exists a Formula such that Formula for all Formula for which Formula.
    5. For every x,Formula, there exists an Formula and Formula such that Formula and Formula.
    6. For every Formula and Formula, there exists a Formula such that Formula for all Formula for which Formula.
    7. For every Formula, there exists a Formula such that Formula for all Formula for which Formula.
  9. Let Formula be a function from a metric space Formula to a metric space Formula. What does it mean for Formula to be uniformly continuous on Formula?
    1. For every Formula and Formula, there exists a Formula such that Formula for all Formula for which Formula.
      • INCORRECT. This is what it means for Formula to be continuous on Formula.
    2. For every Formula, there exists a Formula such that Formula for all Formula for which Formula.
      • CORRECT.
    3. For every Formula, there exists a Formula and Formula such that Formula and Formula.
    4. For every Formula and Formula, there exists a Formula such that Formula for all Formula for which Formula.
    5. For every x,Formula, there exists an Formula and Formula such that Formula and Formula.
    6. For every Formula and Formula, there exists a Formula such that Formula for all Formula for which Formula.
    7. For every Formula, there exists a Formula such that Formula for all Formula for which Formula.
  10. Let Formula be a subset of a metric space Formula. If Formula is both open and closed, then we can conclude that
    1. The boundary of Formula is the empty set.
      • CORRECT.
    2. Formula is either the empty set, or the whole space Formula.
      • INCORRECT. This is a sufficient condition for Formula to be both open and closed, but not necessary (unless Formula is connected).
    3. Formula is the empty set.
    4. Formula is the empty set.
    5. Formula is disconnected.
      • INCORRECT. This is only true if Formula is a non-empty proper subset of Formula.
    6. Formula is disconnected.
    7. Formula has no interior and no exterior.
  11. Let Formula be a subset of a metric space Formula. We say that Formula is complete if and only if
    1. Every Cauchy sequence in Formula, converges in Formula.
      • CORRECT.
    2. Every Cauchy sequence in Formula, converges in Formula.
    3. Every convergent sequence in Formula is Cauchy.
    4. Every sequence in Formula has a subsequence which converges in Formula.
      • INCORRECT. This is what it means for Formula to be compact, not complete.
    5. Every sequence in Formula has a subsequence which converges in Formula.
    6. Every Cauchy sequence in Formula has a convergent subsequence.
    7. Every sequence in Formula has a Cauchy subsequence.
  12. Let Formula be a subset of a metric space Formula. We say that Formula is compact if and only if
    1. Every Cauchy sequence in Formula, converges in Formula.
      • INCORRECT. This is what it means for Formula to be complete, not compact.
    2. Every Cauchy sequence in Formula, converges in Formula.
    3. Every convergent sequence in Formula is Cauchy.
    4. Every sequence in Formula has a subsequence which converges in Formula.
      • CORRECT.
    5. Every sequence in Formula has a subsequence which converges in Formula.
    6. Every Cauchy sequence in Formula has a convergent subsequence.
    7. Every sequence in Formula has a Cauchy subsequence.
  13. Let Formula be a subset of a metric space Formula. We say that Formula is open if and only if
    1. Every point in Formula is an interior point.
      • CORRECT.
    2. Formula contains all of its interior points.
    3. Formula contains all of its adherent points.
      • INCORRECT. This is what it means for Formula to be closed.
    4. Every point in Formula is an adherent point.
    5. Formula has no boundary.
    6. Formula has no exterior.
    7. Formula does not contain any adherent points.
  14. Let Formula be a subset of a metric space Formula. We say that Formula is closed if and only if
    1. Every point in Formula is an interior point.
      • INCORRECT. This is what it means for Formula to be open.
    2. Formula contains all of its interior points.
    3. Formula contains all of its adherent points.
      • CORRECT.
    4. Every point in Formula is an adherent point.
    5. Formula has no boundary.
    6. Formula has no interior.
    7. Formula does not contain any interior points.
  15. Let Formula be a subset of a metric space Formula. We say that Formula is disconnected if and only if
    1. There exist disjoint open sets Formula, Formula in Formula which cover Formula and which both have non-empty intersection with Formula.
      • CORRECT.
    2. There exist disjoint non-empty open sets Formula, Formula in Formula which cover Formula.
    3. There exist disjoint open sets Formula, Formula in Formula which both have non-empty intersection with Formula.
    4. There exist disjoint non-empty open sets Formula, Formula in Formula whose union is Formula.
    5. There exist open sets Formula, Formula in Formula whose union contains Formula, and who both have non-empty intersection with Formula.
    6. Both Formula and Formula are open and non-empty.
    7. Formula can be partitioned into two disjoint non-empty pieces, one of which is open and the other of which is closed.
  16. In the real line Formula with the usual metric, the rationals Formula are
    1. Bounded.
    2. Closed.
    3. Compact.
    4. Complete.
    5. Connected.
    6. Open.
    7. None of the above.
      • CORRECT.
  17. In the real line Formula with the discrete metric, the rationals Formula are
    1. Bounded, Closed, Complete, and Open.
      • CORRECT.
    2. Closed, Compact, Complete and Connected.
    3. Open, Bounded, Connected, and Complete.
    4. Closed, Bounded, Complete, and Compact.
    5. Complete, Connected, Open, and Bounded.
    6. Open, Closed, Bounded, and Compact.
    7. None of the above.

 

Score:  

 

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