Working with Compactness through Covers

1) Which of the following is an open cover of [0,2]?

{ (−1,1), (1,3) }

{ (0,1), (1,2) }

{ (0,2) }

{ [0,2] }

Together, (−1,1) and (1,3) cover [0,2].

Explanation

Together, (−1,1) and (1,3) cover [0,2].

2) A single open interval (−1,3) is an open cover of [0,2].

True

False

One open interval (−1,3) contains [0,2].

Explanation

One open interval (−1,3) contains [0,2].

3) Which is not an open cover of (0,1)?

{ (0,0.5), (0.5,1) }

{ (0,1) }

{ (−1,0), (1,2) }

{ (−0.5,0.5), (0.5,1.5) }

(−1,0) and (1,2) leave out (0,1).

Explanation

(−1,0) and (1,2) leave out (0,1).

4) Which is a finite subcover of { (−1,0.6), (0.4,0.8), (0.7,1.5) } for [0,1]?

{ (−1,0.6), (0.7,1.5) }

{ (−1,0.6) }

{ (0.7,1.5) }

{ (0.4,0.8) }

These two cover the whole [0,1].

Explanation

These two cover the whole [0,1].

5) Every closed interval [a,b] in ℝ is compact.

True

False

Closed intervals are compact by Heine–Borel.

Explanation

Closed intervals are compact by Heine–Borel.

6) Which is an infinite open cover of (0,1)?

{ (0,1) }

{ (1/n,1) : n ∈ ℕ }

{ [0,1] }

{ (−1,2) }

Infinitely many (1/n,1) intervals cover (0,1).

Explanation

Infinitely many (1/n,1) intervals cover (0,1).

7) Which property distinguishes compact sets in ℝ?

Open

Closed and bounded

Infinite

Finite

Compact sets in ℝ are closed and bounded.

Explanation

Compact sets in ℝ are closed and bounded.

8) Which of these intervals is compact?

(−∞,1]

(0,1)

[0,1]

(0,∞)

[0,1] is closed and bounded.

Explanation

[0,1] is closed and bounded.

9) An unbounded set can be compact.

True

False

Compact sets must be bounded.

Explanation

Compact sets must be bounded.

10) Which open cover has no finite subcover for (0,1)?

{ (0,1) }

{ (1/n,1) : n ∈ ℕ }

{ (−1,2) }

{ (0,0.5),(0.5,1) }

(1/n,1) requires infinitely many sets.

Explanation

(1/n,1) requires infinitely many sets.

11) Which interval is not compact?

[0,5]

(0,5)

[−2,2]

{1,2,3}

Open intervals are not compact.

Explanation

Open intervals are not compact.

12) Which is an open cover of [−1,1]?

{ (−2,0), (0,2) }

{ [−1,0], [0,1] }

{ {−1}, {0}, {1} }

{ (−1,1) }

These two open intervals together cover [−1,1].

Explanation

These two open intervals together cover [−1,1].

13) The intersection of two compact sets is compact.

True

False

Intersection of compact sets is compact.

Explanation

Intersection of compact sets is compact.

14) Which is an open cover of the point {0}?

{ (−1,1) }

{ {0} }

{ [0,0] }

None

Any open set containing 0 covers {0}.

Explanation

Any open set containing 0 covers {0}.

15) Which property ensures [0,1] has a finite subcover?

Openness

Closedness and boundedness

Infinite cardinality

Being countable

Closed + bounded → compact.

Explanation

Closed + bounded → compact.