Back
0

Open and Closed Sets

Description: This quiz will test your understanding of open and closed sets in topology.
Number of Questions: 14
Created by:
Tags: topology open sets closed sets
Start the Quiz
Attempted 0/14 Correct 0 Score 0
‹
›

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

Show answer
Correct Option: D
Explanation:

A set is open if it contains all of its limit points. The set of all numbers greater than or equal to 0 is open because it contains all of its limit points, which are the numbers greater than 0.

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

Show answer
Correct Option: D
Explanation:

A set is closed if it contains all of its limit points. The set of all numbers less than or equal to 0 is closed because it contains all of its limit points, which are the numbers less than 0.

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.

Show answer
Correct Option: C
Explanation:

A set is open if and only if its complement is closed, and a set is closed if and only if its complement is open. This is known as the complement rule.

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.

Show answer
Correct Option:
Explanation:

The union of two open sets is always open, the intersection of two open sets is always open, the union of two closed sets is always closed, and the intersection of two closed sets is always closed. These are known as the union and intersection rules.

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.

Show answer
Correct Option: D
Explanation:

If $A$ is open, then its closure is not necessarily open. For example, the set of all rational numbers is open in the real numbers, but its closure, which is the set of all real numbers, is not open. Similarly, if $A$ is closed, then its interior is not necessarily closed. For example, the set of all irrational numbers is closed in the real numbers, but its interior, which is the empty set, is not closed.

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.

Show answer
Correct Option: C
Explanation:

The boundary of an open set is always closed, and the boundary of a closed set is always open. This is known as the boundary rule.

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.

Show answer
Correct Option: C
Explanation:

A set is open if and only if it is the union of open sets, and a set is closed if and only if it is the intersection of closed sets. These are known as the union and intersection theorems.

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.

Show answer
Correct Option: C
Explanation:

A set is open if and only if it contains all of its limit points, and a set is closed if and only if it contains none of its limit points. This is known as the limit point theorem.

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.

Show answer
Correct Option: C
Explanation:

A set is open if and only if its complement is closed, and a set is closed if and only if its complement is open. This is known as the complement rule.

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.

Show answer
Correct Option:
Explanation:

The union of two open sets is always open, the intersection of two open sets is always open, the union of two closed sets is always closed, and the intersection of two closed sets is always closed. These are known as the union and intersection rules.

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.

Show answer
Correct Option: D
Explanation:

If $A$ is open, then its closure is not necessarily open. For example, the set of all rational numbers is open in the real numbers, but its closure, which is the set of all real numbers, is not open. Similarly, if $A$ is closed, then its interior is not necessarily closed. For example, the set of all irrational numbers is closed in the real numbers, but its interior, which is the empty set, is not closed.

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.

Show answer
Correct Option: C
Explanation:

The boundary of an open set is always closed, and the boundary of a closed set is always open. This is known as the boundary rule.

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.

Show answer
Correct Option: C
Explanation:

A set is open if and only if it is the union of open sets, and a set is closed if and only if it is the intersection of closed sets. These are known as the union and intersection theorems.

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.

Show answer
Correct Option: C
Explanation:

A set is open if and only if it contains all of its limit points, and a set is closed if and only if it contains none of its limit points. This is known as the limit point theorem.

- Hide questions
ADVERTISEMENT

Find more quizzes from top tags

general knowledge
3615 Quizzes
119858 Questions
 technology
307 Quizzes
36705 Questions
 sports
963 Quizzes
19148 Questions
 history
1187 Quizzes
13475 Questions
 physics
598 Quizzes
13441 Questions
 biology
1160 Quizzes
12174 Questions
 science & technology
551 Quizzes
11015 Questions
 chemistry
490 Quizzes
9892 Questions
 maths
518 Quizzes
9324 Questions
 softskills
0 Quizzes
7131 Questions