Completeness: Definition Mastery Quiz

1) Every complete metric space is also a closed subset of itself.

True

False

True, because every metric space is closed in itself; completeness does not affect this fact.

Explanation

True, because every metric space is closed in itself; completeness does not affect this fact.

2) Which statement is true about a Cauchy sequence in a metric space?

Every Cauchy sequence is convergent in any metric space

A Cauchy sequence’s terms get arbitrarily close to each other

A Cauchy sequence always diverges in complete metric spaces

Cauchy sequences cannot exist in incomplete spaces

A Cauchy sequence means the terms become arbitrarily close to one another, but they may or may not converge depending on completeness.

Explanation

A Cauchy sequence means the terms become arbitrarily close to one another, but they may or may not converge depending on completeness.

3) The set of rational numbers ℚ with the usual metric d(x,y)=|x−y| is a complete metric space.

True

False

False, because Cauchy sequences of rationals may converge to irrational limits, which are not in ℚ.

Explanation

False, because Cauchy sequences of rationals may converge to irrational limits, which are not in ℚ.

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

ℚ with usual distance

ℝ with usual distance

Open interval (0,1) ⊂ ℝ

The set of integers ℤ with usual distance

ℝ is complete, and ℤ is complete because every Cauchy sequence in ℤ is eventually constant.

Explanation

ℝ is complete, and ℤ is complete because every Cauchy sequence in ℤ is eventually constant.

5) If a metric space is complete, then any closed subset of it is also complete.

True

False

True, because closed subsets contain all their limit points, ensuring Cauchy sequences converge within the subset.

Explanation

True, because closed subsets contain all their limit points, ensuring Cauchy sequences converge within the subset.

6) Let X = (0,1] with the usual metric. Which is true?

X is complete

X is incomplete

Every Cauchy sequence in X converges in X

X contains all its limit points

X is incomplete because Cauchy sequences may converge to 0, which is not in X.

Explanation

X is incomplete because Cauchy sequences may converge to 0, which is not in X.

7) A metric space is complete if and only if every absolutely convergent series of elements converges.

True

False

False, because completeness concerns Cauchy sequences, not series. Absolute convergence is a concept specific to normed spaces.

Explanation

False, because completeness concerns Cauchy sequences, not series. Absolute convergence is a concept specific to normed spaces.

8) Which property guarantees that a metric space (X,d) is complete?

Every bounded sequence converges

Every Cauchy sequence converges in X

Every sequence has a convergent subsequence

The space is finite

Completeness is defined by the convergence of all Cauchy sequences.

Explanation

Completeness is defined by the convergence of all Cauchy sequences.

9) All finite metric spaces are complete.

True

False

True, because any sequence in a finite set must eventually repeat values, making all Cauchy sequences converge.

Explanation

True, because any sequence in a finite set must eventually repeat values, making all Cauchy sequences converge.

10) Consider C[0,1] with the supremum metric d(f,g) = sup |f(x) − g(x)|. Which statement is correct?

C[0,1] is incomplete

C[0,1] is complete

C[0,1] is complete only if functions are differentiable

C[0,1] is complete only for polynomial functions

C[0,1] under the sup metric is a classical example of a complete metric space.

Explanation

C[0,1] under the sup metric is a classical example of a complete metric space.

11) If X is complete and f:X→X is a contraction mapping, then f has a unique fixed point.

True

False

True, by the Banach Fixed Point Theorem, contractions on complete spaces have unique fixed points.

Explanation

True, by the Banach Fixed Point Theorem, contractions on complete spaces have unique fixed points.

12) Which of the following statements is true?

Every complete metric space is bounded

Every bounded metric space is complete

Completeness is independent of boundedness

Completeness implies compactness

Completeness and boundedness are unrelated; a space can be complete yet unbounded (ℝ), or bounded but incomplete.

Explanation

Completeness and boundedness are unrelated; a space can be complete yet unbounded (ℝ), or bounded but incomplete.

13) The space ℓ² of square-summable sequences is a complete metric space.

True

False

True, ℓ² is a Hilbert space and is known to be complete.

Explanation

True, ℓ² is a Hilbert space and is known to be complete.

14) Which subset of ℝ² with the Euclidean metric is complete?

Open unit disk {(x,y): x² + y² < 1}

Closed unit disk {(x,y): x² + y² ≤ 1}

Punctured disk {(x,y): 0 < x² + y² ≤ 1}

Open square (0,1)×(0,1)

The closed disk is closed in a complete space (ℝ²), so it is complete. All other choices exclude boundary points.

Explanation

The closed disk is closed in a complete space (ℝ²), so it is complete. All other choices exclude boundary points.

15) Completeness is preserved under isometric mappings.

True

False

True, because an isometry preserves distances, so convergence and completeness are maintained.

Explanation

True, because an isometry preserves distances, so convergence and completeness are maintained.