Heine–Borel Theorem Quiz

1) In ℝⁿ with the usual topology, a set is compact if and only if it is closed and bounded.

True

False

The Heine–Borel theorem states exactly this for ℝⁿ.

Explanation

The Heine–Borel theorem states exactly this for ℝⁿ.

2) The Heine–Borel Theorem applies to all topological spaces.

True

False

Heine–Borel only holds in Euclidean spaces.

Explanation

Heine–Borel only holds in Euclidean spaces.

3) The interval [0,1] is compact in ℝ because it is closed and bounded.

True

False

Closed + bounded ⇒ compact.

Explanation

Closed + bounded ⇒ compact.

4) The set (0,1) is not compact in ℝ because it is not closed.

True

False

It is bounded but not closed.

Explanation

It is bounded but not closed.

5) A closed interval [a,b] is compact even if a > b.

True

False

If a>b interval empty; empty sets are compact but expression invalid.

Explanation

If a>b interval empty; empty sets are compact but expression invalid.

6) Boundedness alone is sufficient for compactness in ℝ.

True

False

Must also be closed.

Explanation

Must also be closed.

7) Closedness alone is sufficient for compactness in ℝ.

True

False

ℤ is closed but unbounded.

Explanation

ℤ is closed but unbounded.

8) Every compact subset of ℝ must have a finite maximum and minimum value.

True

False

Continuous identity attains max/min.

Explanation

Continuous identity attains max/min.

9) Any finite union of compact subsets of ℝ is compact.

True

False

Finite union compact.

Explanation

Finite union compact.

10) Heine–Borel guarantees every open cover of closed+bounded set has finite subcover.

True

False

Exactly Heine–Borel.

Explanation

Exactly Heine–Borel.

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

(0,1)

[0,1)

[0,1]

(-1,1)

Only [0,1] is closed+bounded.

Explanation

Only [0,1] is closed+bounded.

12) Which set is not bounded and therefore not compact in ℝ?

[-2,2]

(0,1)

[1,∞)

{0,1,2}

[1,∞) unbounded.

Explanation

[1,∞) unbounded.

13) By Heine–Borel, which condition guarantees compactness in ℝⁿ?

Closed only

Bounded only

Closed and bounded

Containing its interior

Closed+bounded.

Explanation

Closed+bounded.

14) Which set is not compact even though closed?

[0,1]

[0,10]

ℤ

[-5,5]

ℤ closed but unbounded.

Explanation

ℤ closed but unbounded.

15) Which set must contain sup & inf because compact?

[0,1]∪(1,2)

Cantor set

(0,1)

(1,2]

Cantor set compact.

Explanation

Cantor set compact.

Heine–Borel Theorem Quiz

1) In ℝⁿ with the usual topology, a set is compact if and only if it is closed and bounded.

2)

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

2) The Heine–Borel Theorem applies to all topological spaces.

3) The interval [0,1] is compact in ℝ because it is closed and bounded.

4) The set (0,1) is not compact in ℝ because it is not closed.

5) A closed interval [a,b] is compact even if a > b.

6) Boundedness alone is sufficient for compactness in ℝ.

7) Closedness alone is sufficient for compactness in ℝ.

8) Every compact subset of ℝ must have a finite maximum and minimum value.

9) Any finite union of compact subsets of ℝ is compact.

10) Heine–Borel guarantees every open cover of closed+bounded set has finite subcover.

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

12) Which set is not bounded and therefore not compact in ℝ?

13) By Heine–Borel, which condition guarantees compactness in ℝⁿ?

14) Which set is not compact even though closed?

15) Which set must contain sup & inf because compact?