Heine–Borel Compactness Concepts Quiz

1) If a set in ℝ is bounded but not closed, it cannot be compact.

True

False

In ℝ, compact ⇔ closed and bounded. Missing closedness ⇒ not compact.

Explanation

In ℝ, compact ⇔ closed and bounded. Missing closedness ⇒ not compact.

2) The open interval (0,1) is not compact because it fails to contain all of its limit points.

True

False

(0,1) is missing 0 and 1, so it is not closed → not compact.

Explanation

(0,1) is missing 0 and 1, so it is not closed → not compact.

3) A finite union of compact sets in ℝ is compact.

True

False

Compactness is preserved under finite unions.

Explanation

Compactness is preserved under finite unions.

4) [0,1] ∪ (2,3) is compact because it is bounded, even though it is not closed.

True

False

[0,1] ∪ (2,3) is not closed, so it is not compact.

Explanation

[0,1] ∪ (2,3) is not closed, so it is not compact.

5) Every open cover of a compact set in ℝ must contain a finite subcover.

True

False

This is the definition of compactness.

Explanation

This is the definition of compactness.

6) Which of the following sets is compact in ℝ?

(−1,1)

[0,1]

[0,1)

(0,1)

Only [0,1] is both closed and bounded.

Explanation

Only [0,1] is both closed and bounded.

7) Which of the following is not compact, even though it is closed?

[0,5]

[−10,10]

ℤ

[2,3]

ℤ is closed but unbounded.

Explanation

ℤ is closed but unbounded.

8) Which set has open cover with no finite subcover?

[0,1]

(0,1)

[−3,5]

Closed disk

(0,1) not compact.

Explanation

(0,1) not compact.

9) A continuous function on compact subset must attain max/min.

Be differentiable

Be injective

Attain max/min

Be monotone

Extreme Value Theorem.

Explanation

Extreme Value Theorem.

10) Condition for compactness in ℝ²?

Boundedness

Closedness

Closed and bounded

Having finite open cover

Heine-Borel.

Explanation

Heine-Borel.

11) Which sets are compact in ℝ?

[0,1]

[0,1) ∪ [2,3]

{1/n}∪{0}

(−1,1)

[0,1] and {1/n}∪{0} are closed and bounded.

Explanation

[0,1] and {1/n}∪{0} are closed and bounded.

Submit

12) Which conditions guarantee not compact?

Unbounded

Not closed

Has open cover w/o finite subcover

Doesn't contain limit pts

All break compactness.

Explanation

All break compactness.

Submit

13) Closed and bounded hence compact?

Cantor set

[a,b]

{0}

(0,1]

(0,1] not closed.

Explanation

(0,1] not closed.

Submit

14) Sequential compact sets?

[−1,1]

(0,1)

{1/n}

Closed ball

Sequential compact ⇔ compact.

Explanation

Sequential compact ⇔ compact.

Submit

15) Noncompact sets via open cover?

(0,1)

(1,∞)

[0,1] ∪ (2,3)

ℝ

All noncompact.

Explanation

All noncompact.

Submit

Heine–Borel Compactness Concepts Quiz

1) If a set in ℝ is bounded but not closed, it cannot be compact.

2)

What first name or nickname would you like us to use?

2) The open interval (0,1) is not compact because it fails to contain all of its limit points.

3) A finite union of compact sets in ℝ is compact.

4) [0,1] ∪ (2,3) is compact because it is bounded, even though it is not closed.

5) Every open cover of a compact set in ℝ must contain a finite subcover.

6) Which of the following sets is compact in ℝ?

7) Which of the following is not compact, even though it is closed?

8) Which set has open cover with no finite subcover?

9) A continuous function on compact subset must attain max/min.

10) Condition for compactness in ℝ²?

11) Which sets are compact in ℝ?

12) Which conditions guarantee not compact?

13) Closed and bounded hence compact?

14) Sequential compact sets?

15) Noncompact sets via open cover?