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Metric Spaces and Topology

Description: This quiz covers fundamental concepts and theorems related to metric spaces and topology.
Number of Questions: 15
Created by:
Tags: metric spaces topology open sets closed sets continuity
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Which of the following is NOT a metric space?

  1. (R, d(x, y) = |x - y|)

  2. (Q, d(x, y) = |x - y|)

  3. (C, d(x, y) = |x - y|)

Show answer
Correct Option: B
Explanation:

The set of rational numbers Q with the Euclidean metric d(x, y) = |x - y| is not a metric space because it violates the triangle inequality.

In a metric space (X, d), a set E is open if:

  1. For each x in E, there exists an r > 0 such that B_r(x) ⊆ E

  2. For each x in E, there exists an r > 0 such that B_r(x) ∩ E = ∅

  3. For each x in E, there exists an r > 0 such that B_r(x) ⊆ X - E

Show answer
Correct Option: A
Explanation:

A set E in a metric space (X, d) is open if for each x in E, there exists an open ball B_r(x) centered at x with radius r such that B_r(x) is entirely contained within E.

In a metric space (X, d), a set E is closed if:

  1. For each x in E, there exists an r > 0 such that B_r(x) ⊆ E

  2. For each x in E, there exists an r > 0 such that B_r(x) ∩ E = ∅

  3. For each x in X - E, there exists an r > 0 such that B_r(x) ∩ E = ∅

Show answer
Correct Option: C
Explanation:

A set E in a metric space (X, d) is closed if for each x in the complement of E (X - E), there exists an open ball B_r(x) centered at x with radius r such that B_r(x) does not intersect E.

Which of the following functions is continuous on the real line R?

  1. f(x) = 1/x

  2. f(x) = |x|

  3. f(x) = sin(x)

Show answer
Correct Option: C
Explanation:

The function f(x) = 1/x is not continuous at x = 0, f(x) = |x| is not continuous at x = 0, but f(x) = sin(x) is continuous on the entire real line.

In a topological space (X, τ), a subset E is compact if:

  1. Every open cover of E has a finite subcover

  2. Every sequence in E has a convergent subsequence

  3. Every continuous function from E to R is bounded

Show answer
Correct Option: A
Explanation:

A subset E of a topological space (X, τ) is compact if every open cover of E has a finite subcover.

Which of the following is a topological property?

  1. Boundedness

  2. Continuity

  3. Compactness

Show answer
Correct Option: C
Explanation:

Compactness is a topological property because it is preserved under homeomorphisms.

In a metric space (X, d), a sequence (x_n) is Cauchy if:

  1. For every ε > 0, there exists an N such that d(x_n, x_m) < ε for all n, m > N

  2. For every ε > 0, there exists an N such that d(x_n, x_N) < ε for all n > N

  3. For every ε > 0, there exists an N such that d(x_N, x_M) < ε for all N, M > n

Show answer
Correct Option: A
Explanation:

A sequence (x_n) in a metric space (X, d) is Cauchy if for every ε > 0, there exists an integer N such that d(x_n, x_m) < ε for all n, m > N.

Which of the following is a complete metric space?

  1. (R, d(x, y) = |x - y|)

  2. (Q, d(x, y) = |x - y|)

  3. (C, d(x, y) = |x - y|)

Show answer
Correct Option: A
Explanation:

The set of real numbers R with the Euclidean metric d(x, y) = |x - y| is a complete metric space, while the set of rational numbers Q and the set of complex numbers C are not complete.

In a topological space (X, τ), a subset E is dense if:

  1. Every non-empty open set in X intersects E

  2. Every point in X is contained in E

  3. Every closed set in X contains E

Show answer
Correct Option: A
Explanation:

A subset E of a topological space (X, τ) is dense if every non-empty open set in X intersects E.

Which of the following is a Hausdorff space?

  1. (R, d(x, y) = |x - y|)

  2. (Q, d(x, y) = |x - y|)

  3. (C, d(x, y) = |x - y|)

Show answer
Correct Option: A
Explanation:

The set of real numbers R with the Euclidean metric d(x, y) = |x - y| is a Hausdorff space, while the set of rational numbers Q and the set of complex numbers C are not Hausdorff.

In a metric space (X, d), a function f: X → Y is continuous at a point x_0 if:

  1. For every ε > 0, there exists a δ > 0 such that d(x, x_0) < δ implies d(f(x), f(x_0)) < ε

  2. For every ε > 0, there exists a δ > 0 such that d(x, x_0) < ε implies d(f(x), f(x_0)) < δ

  3. For every ε > 0, there exists a δ > 0 such that d(f(x), f(x_0)) < δ implies d(x, x_0) < ε

Show answer
Correct Option: A
Explanation:

A function f: X → Y is continuous at a point x_0 in a metric space (X, d) if for every ε > 0, there exists a δ > 0 such that d(x, x_0) < δ implies d(f(x), f(x_0)) < ε.

Which of the following is a connected space?

  1. (R, d(x, y) = |x - y|)

  2. (Q, d(x, y) = |x - y|)

  3. (C, d(x, y) = |x - y|)

Show answer
Correct Option: A
Explanation:

The set of real numbers R with the Euclidean metric d(x, y) = |x - y| is a connected space, while the set of rational numbers Q and the set of complex numbers C are not connected.

In a topological space (X, τ), a subset E is nowhere dense if:

  1. The interior of E is empty

  2. The closure of E is empty

  3. The boundary of E is empty

Show answer
Correct Option: A
Explanation:

A subset E of a topological space (X, τ) is nowhere dense if the interior of E is empty.

Which of the following is a locally compact space?

  1. (R, d(x, y) = |x - y|)

  2. (Q, d(x, y) = |x - y|)

  3. (C, d(x, y) = |x - y|)

Show answer
Correct Option: A
Explanation:

The set of real numbers R with the Euclidean metric d(x, y) = |x - y| is a locally compact space, while the set of rational numbers Q and the set of complex numbers C are not locally compact.

In a metric space (X, d), a sequence (x_n) converges to a point x if:

  1. For every ε > 0, there exists an N such that d(x_n, x) < ε for all n > N

  2. For every ε > 0, there exists an N such that d(x_n, x) > ε for all n > N

  3. For every ε > 0, there exists an N such that d(x_n, x) = ε for all n > N

Show answer
Correct Option: A
Explanation:

A sequence (x_n) in a metric space (X, d) converges to a point x if for every ε > 0, there exists an integer N such that d(x_n, x) < ε for all n > N.

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