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Quiz: Point set topology

Page history last edited by RH 15 years, 4 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.)

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

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