Heine–Borel Applications Quiz

1) The set ([0,1) ∪ (1,2]) is compact in ℝ.

True

False

The set is not closed because it excludes 1, so it cannot be compact in ℝ.

Explanation

The set is not closed because it excludes 1, so it cannot be compact in ℝ.

2) A sequence in a compact subset of ℝ always contains a convergent subsequence.

True

False

Bolzano–Weierstrass theorem.

Explanation

Bolzano–Weierstrass theorem.

3) The Heine–Borel theorem implies that ℝ is not compact.

True

False

ℝ is unbounded.

Explanation

ℝ is unbounded.

4) The set of natural numbers ℕ is compact.

True

False

ℕ is closed but unbounded.

Explanation

ℕ is closed but unbounded.

5) If a subset of ℝ is compact, then it is contained in some closed interval.

True

False

Compact sets are bounded.

Explanation

Compact sets are bounded.

6) Open cover showing (0,1] not compact

{(0,2)}

{(0,1]}

((1/(n+1),2))

((0,1/n))

Does not cover near 0 finitely.

Explanation

Does not cover near 0 finitely.

7) Which subset is compact?

[0,1]∪[2,3]

[0,1)∪[2,3]

(0,1)∪(2,3)

(0,0)

Finite union of compact sets.

Explanation

Finite union of compact sets.

8) Continuous f on compact K attains max/min

True

False

Extreme Value Theorem.

Explanation

Extreme Value Theorem.

9) Which set is compact in ℝ²?

Unit circle

(0,1)²

ℝ²

Open disk

Closed and bounded.

Explanation

Closed and bounded.

10) Which property fails for (0,1]?

Not open

Not closed

Unbounded

Contains 1

Not closed.

Explanation

Not closed.

11) Heine-Borel application

Differentiability

Finite open cover

Contains limit points

Closed+bounded ⇒ compact

Heine–Borel.

Explanation

Heine–Borel.

12) Closed+bounded but not compact

Closed disk

Closed interval

Closed bounded subset of ℓ²

Rectangle

Infinite dimensional.

Explanation

Infinite dimensional.

13) Compact subset of ℝ must be

Infinite

Open & bounded

Closed & bounded

Countable

Heine–Borel.

Explanation

Heine–Borel.

14) Set with open cover w/o finite subcover

[0,1]

[-2,2]

(0,1)

Closed ball

Not compact.

Explanation

Not compact.

15) Compact sets in ℝ

Every closed set

Every bounded set

Closed & bounded

Must be interval

Compact ⇒ closed & bounded.

Explanation

Compact ⇒ closed & bounded.

Heine–Borel Applications Quiz

1) The set ([0,1) ∪ (1,2]) is compact in ℝ.

2)

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

2) A sequence in a compact subset of ℝ always contains a convergent subsequence.

3) The Heine–Borel theorem implies that ℝ is not compact.

4) The set of natural numbers ℕ is compact.

5) If a subset of ℝ is compact, then it is contained in some closed interval.

6) Open cover showing (0,1] not compact

7) Which subset is compact?

8) Continuous f on compact K attains max/min

9) Which set is compact in ℝ²?

10) Which property fails for (0,1]?

11) Heine-Borel application

12) Closed+bounded but not compact

13) Compact subset of ℝ must be

14) Set with open cover w/o finite subcover

15) Compact sets in ℝ