Problem Solving with Open Covers

1) Compactness means:

Set is open

Every open cover has a finite subcover

Set is finite

Set is infinite

Compactness = finite subcover property.

Explanation

Compactness = finite subcover property.

2) [0,1] is compact because it is closed and bounded.

True

False

Heine–Borel guarantees this.

Explanation

Heine–Borel guarantees this.

3) Which is compact in ℝ?

(0,1)

[0,1]

(−∞,∞)

(0,∞)

Only [0,1] is closed and bounded.

Explanation

Only [0,1] is closed and bounded.

4) Which open cover requires a finite subcover to prove compactness?

{ (−1,0.5), (0.5,2) } for [0,1]

{ (0,1) }

{ [0,1] }

{ (−2,2) }

Compactness ensures finite subcover exists.

Explanation

Compactness ensures finite subcover exists.

5) Infinite sets can be compact.

True

False

Example: [0,1].

Explanation

Example: [0,1].

6) Which is not compact?

[0,1]

(0,1)

{1,2,3}

∅

Open intervals are not compact.

Explanation

Open intervals are not compact.

7) Which of the following is an open cover of [0,1]?

{ (0,1) }

{ (−1,0.5), (0.5,2) }

{ [0,1] }

{ (−∞,∞) }

These two intervals cover [0,1].

Explanation

These two intervals cover [0,1].

8) Every compact subset of ℝ is closed.

True

False

Compact sets are closed.

Explanation

Compact sets are closed.

9) Which condition is necessary for compactness in ℝ?

Closedness

Boundedness

Both a and b

Openness

Compact = closed + bounded.

Explanation

Compact = closed + bounded.

10) Compact sets are always:

Open

Closed

Infinite

Countable

Compact subsets of ℝ are closed.

Explanation

Compact subsets of ℝ are closed.

11) Which set is compact?

ℕ

[−5,5]

(0,∞)

(−∞,∞)

[−5,5] is closed and bounded.

Explanation

[−5,5] is closed and bounded.

12) Finite sets are:

Always compact

Never compact

Compact only if bounded

Compact only if infinite

Finite sets are compact by definition.

Explanation

Finite sets are compact by definition.

13) Which is a finite subcover of { (0,0.6),(0.4,1),(0.8,2) } for [0,1]?

{ (0,0.6),(0.8,2) }

{ (0,0.6) }

{ (0.8,2) }

None

These two intervals cover [0,1].

Explanation

These two intervals cover [0,1].

14) Any open interval (a,b) is compact.

True

False

Open intervals are not compact.

Explanation

Open intervals are not compact.

15) Which theorem characterizes compact subsets of ℝⁿ?

Fundamental theorem of algebra

Heine–Borel theorem

Intermediate value theorem

Mean value theorem

Heine–Borel states compact = closed + bounded.

Explanation

Heine–Borel states compact = closed + bounded.