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Open and Closed Sets

This quiz will test your understanding of open and closed sets in topology.

14 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

Which of the following sets is open in the real numbers with the usual topology?

  1. The set of all rational numbers
  2. The set of all irrational numbers
  3. The set of all numbers between 0 and 1
  4. The set of all numbers greater than or equal to 0
Question 2 Multiple Choice (Single Answer)

Which of the following sets is closed in the real numbers with the usual topology?

  1. The set of all rational numbers
  2. The set of all irrational numbers
  3. The set of all numbers between 0 and 1
  4. The set of all numbers less than or equal to 0
Question 3 Multiple Choice (Single Answer)

Let $X$ be a topological space and $A$ be a subset of $X$. Which of the following is true?

  1. $A$ is open if and only if its complement is closed.
  2. $A$ is closed if and only if its complement is open.
  3. Both of the above.
  4. None of the above.
Question 4 Multiple Choice (Single Answer)

Let $X$ be a topological space and $A$ and $B$ be subsets of $X$. Which of the following is true?

  1. The union of two open sets is always open.
  2. The intersection of two open sets is always open.
  3. The union of two closed sets is always closed.
  4. The intersection of two closed sets is always closed.
Question 5 Multiple Choice (Single Answer)

Let $X$ be a topological space and $A$ be a subset of $X$. Which of the following is true?

  1. If $A$ is open, then its closure is also open.
  2. If $A$ is closed, then its interior is also closed.
  3. Both of the above.
  4. None of the above.
Question 6 Multiple Choice (Single Answer)

Let $X$ be a topological space and $A$ be a subset of $X$. Which of the following is true?

  1. The boundary of an open set is always closed.
  2. The boundary of a closed set is always open.
  3. Both of the above.
  4. None of the above.
Question 7 Multiple Choice (Single Answer)

Let $X$ be a topological space and $A$ be a subset of $X$. Which of the following is true?

  1. A set is open if and only if it is the union of open sets.
  2. A set is closed if and only if it is the intersection of closed sets.
  3. Both of the above.
  4. None of the above.
Question 8 Multiple Choice (Single Answer)

Let $X$ be a topological space and $A$ be a subset of $X$. Which of the following is true?

  1. A set is open if and only if it contains all of its limit points.
  2. A set is closed if and only if it contains none of its limit points.
  3. Both of the above.
  4. None of the above.
Question 9 Multiple Choice (Single Answer)

Let $X$ be a topological space and $A$ be a subset of $X$. Which of the following is true?

  1. A set is open if and only if its complement is closed.
  2. A set is closed if and only if its complement is open.
  3. Both of the above.
  4. None of the above.
Question 10 Multiple Choice (Single Answer)

Let $X$ be a topological space and $A$ be a subset of $X$. Which of the following is true?

  1. The union of two open sets is always open.
  2. The intersection of two open sets is always open.
  3. The union of two closed sets is always closed.
  4. The intersection of two closed sets is always closed.
Question 11 Multiple Choice (Single Answer)

Let $X$ be a topological space and $A$ be a subset of $X$. Which of the following is true?

  1. If $A$ is open, then its closure is also open.
  2. If $A$ is closed, then its interior is also closed.
  3. Both of the above.
  4. None of the above.
Question 12 Multiple Choice (Single Answer)

Let $X$ be a topological space and $A$ be a subset of $X$. Which of the following is true?

  1. The boundary of an open set is always closed.
  2. The boundary of a closed set is always open.
  3. Both of the above.
  4. None of the above.
Question 13 Multiple Choice (Single Answer)

Let $X$ be a topological space and $A$ be a subset of $X$. Which of the following is true?

  1. A set is open if and only if it is the union of open sets.
  2. A set is closed if and only if it is the intersection of closed sets.
  3. Both of the above.
  4. None of the above.
Question 14 Multiple Choice (Single Answer)

Let $X$ be a topological space and $A$ be a subset of $X$. Which of the following is true?

  1. A set is open if and only if it contains all of its limit points.
  2. A set is closed if and only if it contains none of its limit points.
  3. Both of the above.
  4. None of the above.
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