Completeness: Foundation Theorem Quiz

1) A metric space is complete if every Cauchy sequence in the space converges to a point in that space.

True

False

True, because this is exactly the definition of completeness.

Explanation

True, because this is exactly the definition of completeness.

2) Which of the following is always true in a complete metric space?

Every bounded sequence converges

Every Cauchy sequence converges

Every convergent sequence is bounded

Every open set is closed

Completeness means exactly that every Cauchy sequence converges in the space.

Explanation

Completeness means exactly that every Cauchy sequence converges in the space.

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

True

False

True, because any Cauchy sequence in a closed subset converges in the whole space and its limit remains in the closed subset.

Explanation

True, because any Cauchy sequence in a closed subset converges in the whole space and its limit remains in the closed subset.

4) Which of the following subsets of ℝ is complete under the usual metric?

(0,1)

[0,1]

ℚ

(1,∞)

[0,1] is closed in ℝ and ℝ is complete, so [0,1] is complete. The others are missing limit points.

Explanation

[0,1] is closed in ℝ and ℝ is complete, so [0,1] is complete. The others are missing limit points.

5) Finite metric spaces are always complete.

True

False

True, because any Cauchy sequence in a finite set must eventually be constant and hence convergent.

Explanation

True, because any Cauchy sequence in a finite set must eventually be constant and hence convergent.

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

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 such as 1/n converge to 0, which is not in X.

Explanation

X is incomplete because Cauchy sequences such as 1/n converge to 0, which is not in X.

7) The Banach Fixed-Point Theorem requires the metric space to be complete.

True

False

True, the theorem guarantees a unique fixed point for a contraction only in complete metric spaces.

Explanation

True, the theorem guarantees a unique fixed point for a contraction only in complete metric spaces.

8) Which of the following is a necessary and sufficient condition for a metric space (X,d) to be complete?

Every sequence has a convergent subsequence

Every Cauchy sequence converges in X

Every bounded sequence converges

Every closed set contains its limit points

Completeness is defined by the convergence of all Cauchy sequences in the space.

Explanation

Completeness is defined by the convergence of all Cauchy sequences in the space.

9) The set of rational numbers ℚ with the usual metric is a complete metric space.

True

False

False, because some Cauchy sequences of rationals converge to irrationals, which are not in ℚ.

Explanation

False, because some Cauchy sequences of rationals converge to irrationals, which are not in ℚ.

10) Which of the following spaces is complete under the standard metric?

ℚ

(0,∞)

ℝ

(0,1)

ℝ with the usual metric is complete; the other sets are missing some of their Cauchy limits.

Explanation

ℝ with the usual metric is complete; the other sets are missing some of their Cauchy limits.

11) If a metric space is complete, then it is necessarily compact.

True

False

False, completeness does not imply compactness (e.g., ℝ is complete but not compact).

Explanation

False, completeness does not imply compactness (e.g., ℝ is complete but not compact).

12) Consider C[0,1], the space of continuous functions on [0,1] with the sup metric. Which of the following is true?

C[0,1] is incomplete

C[0,1] is complete

Completeness depends on differentiability of functions

Completeness depends on boundedness

C[0,1] with the supremum metric is a classical example of a complete function space.

Explanation

C[0,1] with the supremum metric is a classical example of a complete function space.

13) Every closed and bounded subset of ℝⁿ is complete.

True

False

True, because ℝⁿ is complete and any closed subset of a complete space is complete; boundedness here is extra but still satisfied.

Explanation

True, because ℝⁿ is complete and any closed subset of a complete space is complete; boundedness here is extra but still satisfied.

14) Which of the following is not necessarily complete?

ℝⁿ

Closed interval [0,1]

Open interval (0,1)

ℓ² (space of square-summable sequences)

(0,1) is not complete since Cauchy sequences can converge to 0 or 1, which are not in the interval.

Explanation

(0,1) is not complete since Cauchy sequences can converge to 0 or 1, which are not in the interval.

15) Completeness is preserved under isometric mappings.

True

False

True, if two spaces are isometric (distance-preserving bijection), then Cauchy sequences and their convergence behavior are preserved, so completeness is preserved.

Explanation

True, if two spaces are isometric (distance-preserving bijection), then Cauchy sequences and their convergence behavior are preserved, so completeness is preserved.