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Metric Spaces and Topology

This quiz covers fundamental concepts and theorems related to metric spaces and topology.

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

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|)
Question 2 Multiple Choice (Single Answer)

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
Question 3 Multiple Choice (Single Answer)

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 = ∅
Question 4 Multiple Choice (Single Answer)

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)
Question 5 Multiple Choice (Single Answer)

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
Question 6 Multiple Choice (Single Answer)

Which of the following is a topological property?

  1. Boundedness
  2. Continuity
  3. Compactness
Question 7 Multiple Choice (Single Answer)

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
Question 8 Multiple Choice (Single Answer)

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|)
Question 9 Multiple Choice (Single Answer)

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
Question 10 Multiple Choice (Single Answer)

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|)
Question 11 Multiple Choice (Single Answer)

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
Question 12 Multiple Choice (Single Answer)

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|)
Question 13 Multiple Choice (Single Answer)

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
Question 14 Multiple Choice (Single Answer)

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|)
Question 15 Multiple Choice (Single Answer)

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
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