Compactness in Real Analysis: Definition, Theorems, and Applications

Question 1

In a metric space, a set is compact if:

Accepted Answer

Every open cover has a finite subcover.

Answer

It is closed and bounded.

Answer

It has a finite number of elements.

Answer

It is contained in a larger compact set.

Question 2

The Heine-Borel theorem states that:

Accepted Answer

A closed and bounded subset of a Euclidean space is compact.

Answer

Any sequence in a compact metric space has a convergent subsequence.

Answer

A continuous function on a compact set is uniformly continuous.

Answer

A compact set is dense in any larger set.

Question 3

The Bolzano-Weierstrass theorem states that:

Accepted Answer

Any sequence in a compact metric space has a convergent subsequence.

Answer

A closed and bounded subset of a Euclidean space is compact.

Answer

A continuous function on a compact set is uniformly continuous.

Answer

A compact set is dense in any larger set.

Question 4

Which property is NOT characteristic of compact sets in a metric space?

Accepted Answer

Compact sets are connected.

Answer

Compact sets are closed.

Answer

Compact sets are totally bounded.

Question 5

In a metric space (X, d), which of the following properties is true for compact sets?

Accepted Answer

Closed and bounded

Answer

Closed

Answer

Bounded

Answer

Connected

Question 6

In a metric space (X, d), which statement is equivalent to the compactness of a subset A of X?

Accepted Answer

Every open cover of A has a finite subcover.

Answer

A is both bounded and closed.

Answer

A is totally bounded.

Answer

A is path-connected.

Question 7

The Bolzano-Weierstrass Theorem, in the context of a metric space, asserts that:

Accepted Answer

Every bounded sequence harbors a convergent subsequence.

Answer

Every convergent sequence is inherently bounded.

Answer

Every Cauchy sequence must converge.

Answer

Compact sets are invariably bounded.

Question 8

For a metric space (X, d), which of the following conditions is a sufficient indicator of compactness?

Accepted Answer

Every open covering of X can be reduced to a finite subcovering.

Answer

Every sequence in X yields a convergent subsequence.

Answer

X is bounded and complete.

Answer

X is both connected and path-connected.

Question 9

If a set is closed and bounded in a metric space, then it is:

Accepted Answer

Compact

Answer

Open

Answer

Neither compact nor open

Answer

Possibly compact or open

Question 10

Which theorem states that every bounded sequence in a metric space has a convergent subsequence?

Accepted Answer

Bolzano-Weierstrass Theorem

Answer

Heine-Borel Theorem

Answer

Cauchy-Schwarz Inequality

Answer

Triangle Inequality

Question 11

Consider the set S = {x ∈ ℝ | 0 ≤ x ≤ 1} in the metric space (ℝ, d), where d(x, y) = |x - y|. Is S compact?

Accepted Answer

Yes

Answer

No

Answer

Cannot be determined from the given information

Question 12

Let X be a metric space and K ⊂ X. Which of the following is a sufficient condition for K to be compact?

Accepted Answer

K is closed and totally bounded

Answer

K is open and bounded

Answer

K is closed and connected

Answer

K is open and totally bounded

Question 13

Which of the following is true about the set of rational numbers in the metric space (ℝ, d), where d(x, y) = |x - y|?

Accepted Answer

It is closed but not compact

Answer

It is compact

Answer

It is open

Answer

It is neither open nor closed

Question 14

Which of the following is NOT a property of a compact set?

Accepted Answer

It is complete.

Answer

It is totally bounded.

Answer

It is precompact.

Answer

It is closed.

Question 15

Which of the following statements is equivalent to the Heine-Borel theorem for metric spaces?

Accepted Answer

Every open cover of a compact set in a metric space has a finite subcover.

Answer

Every closed and bounded set in a metric space is compact.

Answer

Every compact set in a metric space is closed and bounded.

Answer

Every bounded set in a metric space is compact.

Question 16

What is the primary application of the Bolzano-Weierstrass theorem?

Accepted Answer

To prove the existence of extreme values for continuous functions on compact sets.

Answer

To demonstrate the existence of fixed points for continuous functions on compact sets.

Answer

To establish the compactness of closed and bounded sets in metric spaces.

Answer

To demonstrate the continuity of differentiable functions.

Question 17

Which property accurately describes any open cover of a compact set in a metric space?

Accepted Answer

It possesses a finite subcover.

Answer

It contains an infinite subcover.

Answer

It has a countable subcover.

Answer

It includes an uncountable subcover.

Question 18

Consider the set S = {x in R : 0 ≤ x ≤ 1}. Determine if S is compact under the usual Euclidean metric.

Accepted Answer

Yes, S is compact because it is both closed and bounded.

Answer

Yes, S is compact because it is open and bounded.

Answer

No, S is not compact because it is unbounded.

Answer

No, S is not compact because it is not closed.

Question 19

In the discrete metric, which set is compact?

Accepted Answer

The set of all natural numbers.

Answer

The set of all rational numbers in the interval [0, 1].

Answer

The set of all irrational numbers in the interval [0, 1].

Answer

The set of all real numbers.

Question 20

Identify an example of a non-compact set in R^2.

Accepted Answer

The set of all points in the first quadrant.

Answer

The closed unit disk.

Answer

The open unit disk.

Answer

The set of all points on the unit circle.

Question 21

Which condition is necessary for a subset of a metric space to be compact?

Accepted Answer

It is closed and bounded.

Answer

It is open and bounded.

Answer

It is closed and unbounded.

Answer

It is open and unbounded.

Question 22

In the Heine-Borel theorem, which condition ensures that a set is compact?

Accepted Answer

It is closed and bounded.

Answer

It is open and bounded.

Answer

It is closed and open.

Answer

It is closed or bounded.

Question 23

Which statement is a consequence of the Bolzano-Weierstrass theorem?

Accepted Answer

Any bounded sequence in a metric space has a convergent subsequence.

Answer

Any convergent sequence in a metric space is bounded.

Answer

Any closed subset of a metric space is compact.

Answer

Any compact subset of a metric space is closed.

Question 24

Consider the set A = {1, 2, 3} in the real numbers with the usual metric. Which statement is true about A?

Accepted Answer

A is both compact and closed.

Answer

A is compact but not closed.

Answer

A is closed but not compact.

Answer

A is neither compact nor closed.

Question 25

Let (X, d) be a metric space and let A ⊆ X. If A is compact and B ⊆ A, then which statement is true?

Accepted Answer

B is also compact.

Answer

B is not necessarily compact.

Answer

B is open.

Answer

B is closed.

Question 26

Consider the following statement: 'If a metric space (X, d) is compact, then every continuous function f: X → R is bounded.' Is this statement true or false?

Accepted Answer

True

Answer

False

Question 27

Which of the following sets provides a counterexample to the converse of the statement in the previous question?

Accepted Answer

The set of all continuous functions on the interval [0, 1].

Answer

The set of rational numbers in the interval [0, 1].

Answer

The set of all differentiable functions on the interval [0, 1].

Answer

The set of all polynomials with real coefficients.

Question 28

Which of the following is equivalent to the Heine-Borel Theorem?

Accepted Answer

Every open covering of a compact set in a metric space has a finite subcovering.

Answer

Every open covering of a closed interval has a finite subcovering.

Answer

Every sequence in a bounded set in a metric space has a convergent subsequence.

Answer

Every continuous function on a closed and bounded interval is uniformly continuous.

Question 29

In the context of metric spaces, which of the following properties characterizes compactness?

Accepted Answer

Closed and totally bounded.

Answer

Closed and bounded.

Answer

Open and bounded.

Answer

Open and totally bounded.

Question 30

Consider the sequence {x_n} = 1/n in the metric space of rational numbers with the usual metric. Does this sequence satisfy the Bolzano-Weierstrass Theorem?

Accepted Answer

No, because it does not have a convergent subsequence.

Answer

Yes, because it is bounded and has a convergent subsequence.

Answer

No, because it is not bounded.

Answer

Yes, because it is bounded and does not have a convergent subsequence.

Question 31

Let f(x) be a continuous function defined on the compact set [a, b]. Which of the following statements is true?

Accepted Answer

f(x) attains both its maximum and minimum values on [a, b].

Answer

f(x) is differentiable on [a, b].

Answer

f(x) is increasing on [a, b].

Answer

f(x) is constant on [a, b].

Question 32

Which of the following is not a property of compact sets in metric spaces?

Accepted Answer

They can be covered by an arbitrary collection of open sets.

Answer

They are closed.

Answer

They are bounded.

Answer

They are totally bounded.

Question 33

In a metric space, which condition is a necessary requirement for compactness?

Accepted Answer

Closed and bounded.

Answer

Open and unbounded.

Answer

Closed and unbounded.

Answer

Open and bounded.

Question 34

Among the following sets in the Euclidean metric, which one is not precompact?

Accepted Answer

The set of integers.

Answer

The set of rational numbers.

Answer

The set of all vectors in R^2 with norm less than 1.

Answer

The set of all polynomials with rational coefficients.

Question 35

Let K be a compact set in a metric space X. Which of the following statements about sequences in K is true?

Accepted Answer

Every sequence in K has a subsequence that converges to a point in K.

Answer

Every sequence in X with infinitely many points in K converges to a point in K.

Answer

Every bounded sequence in X with infinitely many points in K converges to a point in K.

Answer

Every convergent sequence in X with infinitely many points in K converges to a point in K.

Question 36

Which of the following is a direct consequence of the Heine-Borel Theorem?

Accepted Answer

Every open cover of a compact set has a finite subcover.

Answer

Every continuous function defined on a compact set is uniformly continuous.

Answer

Every Cauchy sequence in a compact set converges.

Answer

Every bounded sequence in a compact set has a convergent subsequence.

Question 37

The Heine-Borel theorem states that every closed interval in the real numbers is what?

Accepted Answer

Compact.

Answer

Neither compact nor bounded.

Answer

Bounded.

Answer

Open.

Question 38

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

Accepted Answer

Every open cover of S has a finite subcover.

Answer

S is a Cauchy sequence.

Answer

S has a finite number of elements.

Answer

S is closed and bounded.

Question 39

Let S be a subset of a metric space (X, d). If S is compact, then which of the following is also true?

Accepted Answer

Every subset of S is also compact.

Answer

S is open in X.

Answer

S is bounded but not compact.

Answer

S is neither compact nor bounded.

Question 40

Which of the following is a consequence of the compactness of the unit interval [0, 1]?

Accepted Answer

Every continuous function on [0, 1] attains both its minimum and maximum values.

Answer

Every differentiable function on [0, 1] has a critical point.

Answer

Every polynomial function on [0, 1] has a real root.

Question 41

Consider the set S = {x ∈ ℝ : 0 ≤ x ≤ 1} in the metric space (ℝ, d), where d(x, y) = |x - y|. Is S precompact?

Accepted Answer

Yes, as it is totally bounded.

Answer

Yes, as it is compact.

Answer

No, as it is not closed.

Answer

No, as it is not bounded.

Question 42

Let (X, d) be a metric space and S ⊆ X. If S is sequentially compact and closed, then which of the following must be true?

Accepted Answer

S is compact.

Answer

S is totally bounded.

Answer

S is precompact.

Question 43

Is the set of all rational numbers between 0 and 1 (inclusive) compact in the standard metric space of real numbers?

Accepted Answer

No

Answer

Yes, it is bounded and closed.

Answer

Yes, it is a closed set.

Question 44

Which of the following statements is equivalent to the Heine-Borel Theorem for a metric space?

Accepted Answer

Every infinite sequence in a compact metric space has a convergent subsequence.

Answer

Every Cauchy sequence in a compact metric space converges.

Answer

Every bounded sequence in a compact metric space has a convergent subsequence.

Question 45

Let A be a totally bounded subset of a metric space X. Which of the following conditions guarantees that A is compact?

Accepted Answer

A is closed.

Answer

A is a bounded set.

Answer

X is a compact space.

Answer

A is an open set.

Question 46

If A is a compact subset of a metric space X, and f: X → ℝ is a continuous function, is f necessarily bounded on A?

Accepted Answer

Yes

Answer

Yes, only if f is a continuous function.

Answer

No.

Answer

Yes, only if X is complete.

Question 47

Consider the set of all continuous functions on the interval [0, 1] with the metric d(f, g) = sup{|f(x) - g(x)| : x ∈ [0, 1]}. Is this set compact?

Accepted Answer

No

Answer

Yes, it is totally bounded and closed.

Answer

Yes, it is a subset of a compact space.

Answer

Yes, it is bounded and closed.

Question 48

What is a consequence of the compactness of the unit ball in a finite-dimensional Euclidean space?

Accepted Answer

Every bounded sequence in the space has a convergent subsequence.

Answer

The space is complete.

Answer

Every Cauchy sequence in the space converges.

Question 49

Consider the set of all polynomials with real coefficients equipped with the metric d(p, q) = max{|p(x) - q(x)| : x ∈ [0, 1]}. Is this set compact?

Accepted Answer

No

Answer

Yes, it is bounded and closed.

Answer

Yes, it is a subset of a compact space.

Answer

Yes, it is totally bounded and closed.

Question 50

Which of the following sets is NOT compact in the standard metric space of real numbers?

Accepted Answer

(0, 1)

Answer

[0, 1]

Answer

{1, 2, 3}

Answer

[-1, 1]

Question 51

According to the Heine-Borel Theorem, a subset of ℝ^n is compact if and only if it is...

Accepted Answer

Closed and bounded

Answer

Closed and unbounded

Answer

Open and bounded

Answer

Open and unbounded

Question 52

Consider the sequence (xn) = (1/n) in the metric space ℝ. What can be said about its limit points?

Accepted Answer

The sequence has a unique limit point, 0.

Answer

The sequence has infinitely many limit points.

Answer

The sequence has two limit points: 0 and 1.

Answer

The sequence has no limit points.

Question 53

Which of the following statements is TRUE regarding continuous functions on compact sets?

Accepted Answer

A continuous function on a compact set is bounded.

Answer

A continuous function on a compact set is always surjective.

Answer

A continuous function on a compact set is always differentiable.

Answer

A continuous function on a compact set is always injective.

Question 54

Let f be a continuous function from a compact set K to ℝ. Which of the following is always true about the image f(K)?

Accepted Answer

f(K) is compact.

Answer

f(K) is open.

Answer

f(K) is unbounded.

Answer

f(K) is empty.

Question 55

Consider the set A = {1/n | n ∈ ℕ} in the metric space ℝ. Is A compact?

Accepted Answer

No, A is not compact because it is not closed.

Answer

Yes, A is compact because it is closed.

Answer

No, A is not compact because it is unbounded.

Answer

Yes, A is compact because it is bounded.

Question 56

Which of the following is NOT a consequence of the Bolzano-Weierstrass Theorem?

Accepted Answer

Every bounded sequence in a metric space has a convergent subsequence.

Answer

Every infinite sequence in a compact set has a convergent subsequence.

Answer

Every infinite sequence in a compact set has a limit point.

Question 57

A sequence in a metric space is said to be Cauchy if...

Accepted Answer

For any ε > 0, there exists an N such that d(xn, xm) < ε for all n, m ≥ N.

Answer

For any ε > 0, there exists an N such that d(xn, xn+1) < ε for all n ≥ N.

Answer

The sequence converges to a limit point.

Question 58

In a complete metric space, every Cauchy sequence...

Accepted Answer

Converges to a point in the space.

Answer

Is bounded.

Answer

Diverges to infinity.

Answer

Has a subsequence that converges.

Question 59

Which of the following statements is FALSE regarding compact metric spaces?

Accepted Answer

Every compact metric space is complete.

Answer

Every compact metric space is bounded.

Answer

Every continuous function from a compact metric space to another metric space is uniformly continuous.

Answer

Every closed subset of a compact metric space is compact.