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Metric Spaces

Description: This quiz is designed to assess your understanding of the fundamental concepts and properties of metric spaces. It covers topics such as distance functions, open and closed sets, convergence, completeness, and compactness.
Number of Questions: 14
Created by:
Tags: metric spaces topology analysis
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Let (X, d) be a metric space. Which of the following statements is true?

  1. The distance between any two points in X is always positive.

  2. The distance between any two points in X is always non-negative.

  3. The distance between any two points in X is always zero.

  4. The distance between any two points in X is always negative.

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Correct Option: B
Explanation:

In a metric space, the distance between any two points is always non-negative. This is one of the fundamental properties of a metric space.

Let (X, d) be a metric space. A set S in X is said to be open if:

  1. For every x in S, there exists an open ball centered at x that is contained in S.

  2. For every x in S, there exists a closed ball centered at x that is contained in S.

  3. For every x in S, there exists a neighborhood of x that is contained in S.

  4. For every x in S, there exists a boundary point of S that is contained in S.

Show answer
Correct Option: A
Explanation:

In a metric space, a set S is said to be open if for every point x in S, there exists an open ball centered at x that is contained in S.

Let (X, d) be a metric space. A sequence (x_n) in X is said to be convergent if:

  1. There exists a point x in X such that lim_(n->∞) d(x_n, x) = 0.

  2. There exists a point x in X such that lim_(n->∞) d(x_n, x) = ∞.

  3. There exists a point x in X such that lim_(n->∞) d(x_n, x) = 1.

  4. There exists a point x in X such that lim_(n->∞) d(x_n, x) = -1.

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Correct Option: A
Explanation:

In a metric space, a sequence (x_n) is said to be convergent if there exists a point x in X such that lim_(n->∞) d(x_n, x) = 0.

Let (X, d) be a metric space. A set S in X is said to be closed if:

  1. Its complement is open.

  2. Its interior is empty.

  3. Its boundary is empty.

  4. Its closure is empty.

Show answer
Correct Option: A
Explanation:

In a metric space, a set S is said to be closed if its complement is open.

Let (X, d) be a metric space. A metric space is said to be complete if:

  1. Every Cauchy sequence in X converges to a point in X.

  2. Every convergent sequence in X converges to a point in X.

  3. Every bounded sequence in X converges to a point in X.

  4. Every open set in X is dense in X.

Show answer
Correct Option: A
Explanation:

In a metric space, a metric space is said to be complete if every Cauchy sequence in X converges to a point in X.

Let (X, d) be a metric space. A metric space is said to be compact if:

  1. Every open cover of X has a finite subcover.

  2. Every closed cover of X has a finite subcover.

  3. Every sequence in X has a convergent subsequence.

  4. Every bounded sequence in X has a convergent subsequence.

Show answer
Correct Option: A
Explanation:

In a metric space, a metric space is said to be compact if every open cover of X has a finite subcover.

Which of the following is an example of a complete metric space?

  1. The set of real numbers with the usual metric.

  2. The set of rational numbers with the usual metric.

  3. The set of integers with the usual metric.

  4. The set of complex numbers with the usual metric.

Show answer
Correct Option: A
Explanation:

The set of real numbers with the usual metric is an example of a complete metric space.

Which of the following is an example of a compact metric space?

  1. The closed interval [0, 1] with the usual metric.

  2. The open interval (0, 1) with the usual metric.

  3. The set of integers with the usual metric.

  4. The set of rational numbers with the usual metric.

Show answer
Correct Option: A
Explanation:

The closed interval [0, 1] with the usual metric is an example of a compact metric space.

Let (X, d) be a metric space. Which of the following is true about the diameter of a set S in X?

  1. The diameter of S is the supremum of the distances between all pairs of points in S.

  2. The diameter of S is the infimum of the distances between all pairs of points in S.

  3. The diameter of S is the average of the distances between all pairs of points in S.

  4. The diameter of S is the median of the distances between all pairs of points in S.

Show answer
Correct Option: A
Explanation:

The diameter of a set S in a metric space (X, d) is the supremum of the distances between all pairs of points in S.

Let (X, d) be a metric space. Which of the following is true about the boundary of a set S in X?

  1. The boundary of S is the set of all points in X that are in the closure of S but not in the interior of S.

  2. The boundary of S is the set of all points in X that are in the interior of S but not in the closure of S.

  3. The boundary of S is the set of all points in X that are in both the closure of S and the interior of S.

  4. The boundary of S is the set of all points in X that are in neither the closure of S nor the interior of S.

Show answer
Correct Option: A
Explanation:

The boundary of a set S in a metric space (X, d) is the set of all points in X that are in the closure of S but not in the interior of S.

Let (X, d) be a metric space. Which of the following is true about the closure of a set S in X?

  1. The closure of S is the set of all points in X that are in S or in the boundary of S.

  2. The closure of S is the set of all points in X that are in S or in the interior of S.

  3. The closure of S is the set of all points in X that are in both S and the boundary of S.

  4. The closure of S is the set of all points in X that are in neither S nor the boundary of S.

Show answer
Correct Option: A
Explanation:

The closure of a set S in a metric space (X, d) is the set of all points in X that are in S or in the boundary of S.

Let (X, d) be a metric space. Which of the following is true about the interior of a set S in X?

  1. The interior of S is the set of all points in X that are in S but not in the boundary of S.

  2. The interior of S is the set of all points in X that are in S but not in the closure of S.

  3. The interior of S is the set of all points in X that are in both S and the boundary of S.

  4. The interior of S is the set of all points in X that are in neither S nor the boundary of S.

Show answer
Correct Option: A
Explanation:

The interior of a set S in a metric space (X, d) is the set of all points in X that are in S but not in the boundary of S.

Let (X, d) be a metric space. Which of the following is true about the distance between two sets A and B in X?

  1. The distance between A and B is the infimum of the distances between all pairs of points in A and B.

  2. The distance between A and B is the supremum of the distances between all pairs of points in A and B.

  3. The distance between A and B is the average of the distances between all pairs of points in A and B.

  4. The distance between A and B is the median of the distances between all pairs of points in A and B.

Show answer
Correct Option: A
Explanation:

The distance between two sets A and B in a metric space (X, d) is the infimum of the distances between all pairs of points in A and B.

Let (X, d) be a metric space. Which of the following is true about the Hausdorff distance between two sets A and B in X?

  1. The Hausdorff distance between A and B is the supremum of the distances between all points in A and their nearest points in B.

  2. The Hausdorff distance between A and B is the infimum of the distances between all points in A and their nearest points in B.

  3. The Hausdorff distance between A and B is the average of the distances between all points in A and their nearest points in B.

  4. The Hausdorff distance between A and B is the median of the distances between all points in A and their nearest points in B.

Show answer
Correct Option: A
Explanation:

The Hausdorff distance between two sets A and B in a metric space (X, d) is the supremum of the distances between all points in A and their nearest points in B.

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