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Descriptive Set Theory

Descriptive Set Theory Quiz

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Questions

Question 1 Multiple Choice (Single Answer)

What is the Borel hierarchy?

  1. A hierarchy of sets that are defined using transfinite induction
  2. A hierarchy of sets that are defined using the axiom of choice
  3. A hierarchy of sets that are defined using the continuum hypothesis
  4. A hierarchy of sets that are defined using the well-ordering theorem
Question 2 Multiple Choice (Single Answer)

What is the difference between an analytic set and a coanalytic set?

  1. An analytic set is a set that can be defined using a formula in the language of set theory, while a coanalytic set is a set that can be defined using a formula in the language of set theory with the addition of the axiom of choice
  2. An analytic set is a set that can be defined using a formula in the language of set theory, while a coanalytic set is a set that can be defined using a formula in the language of set theory with the addition of the continuum hypothesis
  3. An analytic set is a set that can be defined using a formula in the language of set theory, while a coanalytic set is a set that can be defined using a formula in the language of set theory with the addition of the well-ordering theorem
  4. An analytic set is a set that can be defined using a formula in the language of set theory, while a coanalytic set is a set that can be defined using a formula in the language of set theory with the addition of the axiom of determinacy
Question 3 Multiple Choice (Single Answer)

What is the Lusin separation theorem?

  1. A theorem that states that every analytic set can be separated from every coanalytic set by a Borel set
  2. A theorem that states that every analytic set can be separated from every coanalytic set by a projective set
  3. A theorem that states that every analytic set can be separated from every coanalytic set by a Suslin set
  4. A theorem that states that every analytic set can be separated from every coanalytic set by a Lebesgue measurable set
Question 4 Multiple Choice (Single Answer)

What is the Baire category theorem?

  1. A theorem that states that every complete metric space is a Baire space
  2. A theorem that states that every complete metric space is a Polish space
  3. A theorem that states that every complete metric space is a Suslin space
  4. A theorem that states that every complete metric space is a Lebesgue measurable space
Question 5 Multiple Choice (Single Answer)

What is the perfect set theorem?

  1. A theorem that states that every non-empty perfect set in a complete metric space is uncountable
  2. A theorem that states that every non-empty perfect set in a complete metric space is Lebesgue measurable
  3. A theorem that states that every non-empty perfect set in a complete metric space is a Baire space
  4. A theorem that states that every non-empty perfect set in a complete metric space is a Polish space
Question 6 Multiple Choice (Single Answer)

What is the Cantor-Bendixson theorem?

  1. A theorem that states that every set of real numbers can be decomposed into a perfect set and a countable set
  2. A theorem that states that every set of real numbers can be decomposed into a perfect set and a Lebesgue measurable set
  3. A theorem that states that every set of real numbers can be decomposed into a perfect set and a Baire space
  4. A theorem that states that every set of real numbers can be decomposed into a perfect set and a Polish space
Question 7 Multiple Choice (Single Answer)

What is the Sierpiński-Zygmund theorem?

  1. A theorem that states that every set of real numbers can be decomposed into a perfect set and a set of Lebesgue measure zero
  2. A theorem that states that every set of real numbers can be decomposed into a perfect set and a set of Baire measure zero
  3. A theorem that states that every set of real numbers can be decomposed into a perfect set and a set of Hausdorff measure zero
  4. A theorem that states that every set of real numbers can be decomposed into a perfect set and a set of Carathéodory measure zero
Question 8 Multiple Choice (Single Answer)

What is the Lusin-Novikov theorem?

  1. A theorem that states that every analytic set is Lebesgue measurable
  2. A theorem that states that every analytic set is Baire measurable
  3. A theorem that states that every analytic set is Hausdorff measurable
  4. A theorem that states that every analytic set is Carathéodory measurable
Question 9 Multiple Choice (Single Answer)

What is the Suslin theorem?

  1. A theorem that states that every analytic set is a Suslin set
  2. A theorem that states that every analytic set is a Baire set
  3. A theorem that states that every analytic set is a Polish set
  4. A theorem that states that every analytic set is a Lebesgue measurable set
Question 10 Multiple Choice (Single Answer)

What is the Kuratowski-Ulam theorem?

  1. A theorem that states that every Suslin set is a Polish set
  2. A theorem that states that every Suslin set is a Baire set
  3. A theorem that states that every Suslin set is an analytic set
  4. A theorem that states that every Suslin set is a Lebesgue measurable set
Question 11 Multiple Choice (Single Answer)

What is the Solovay theorem?

  1. A theorem that states that every Lebesgue measurable set is a Suslin set
  2. A theorem that states that every Lebesgue measurable set is a Baire set
  3. A theorem that states that every Lebesgue measurable set is an analytic set
  4. A theorem that states that every Lebesgue measurable set is a Polish set
Question 12 Multiple Choice (Single Answer)

What is the Martin's axiom?

  1. An axiom that states that every family of sets of real numbers with the property that every two sets in the family have a non-empty intersection has a common element
  2. An axiom that states that every family of sets of real numbers with the property that every two sets in the family have a non-empty intersection has a countable intersection
  3. An axiom that states that every family of sets of real numbers with the property that every two sets in the family have a non-empty intersection has a perfect intersection
  4. An axiom that states that every family of sets of real numbers with the property that every two sets in the family have a non-empty intersection has a Lebesgue measurable intersection
Question 13 Multiple Choice (Single Answer)

What is the axiom of determinacy?

  1. An axiom that states that every game of perfect information has a winning strategy
  2. An axiom that states that every game of imperfect information has a winning strategy
  3. An axiom that states that every game of chance has a winning strategy
  4. An axiom that states that every game of skill has a winning strategy
Question 14 Multiple Choice (Single Answer)

What is the continuum hypothesis?

  1. A hypothesis that states that there is no set whose cardinality is greater than the cardinality of the set of natural numbers and less than the cardinality of the set of real numbers
  2. A hypothesis that states that there is a set whose cardinality is greater than the cardinality of the set of natural numbers and less than the cardinality of the set of real numbers
  3. A hypothesis that states that there is a set whose cardinality is equal to the cardinality of the set of natural numbers
  4. A hypothesis that states that there is a set whose cardinality is equal to the cardinality of the set of real numbers
Question 15 Multiple Choice (Single Answer)

What is the generalized continuum hypothesis?

  1. A hypothesis that states that for every set of real numbers, there is a set whose cardinality is greater than the cardinality of the set of natural numbers and less than the cardinality of the set of real numbers
  2. A hypothesis that states that for every set of real numbers, there is a set whose cardinality is equal to the cardinality of the set of natural numbers
  3. A hypothesis that states that for every set of real numbers, there is a set whose cardinality is equal to the cardinality of the set of real numbers
  4. A hypothesis that states that for every set of real numbers, there is a set whose cardinality is greater than the cardinality of the set of natural numbers and greater than the cardinality of the set of real numbers
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