Finite Subcovers Application Quiz

1) Which family is an open cover of (0,1) that has no finite subcover?

{ (−1,2) }

( (1/(n+1), 1 - 1/(n+1)) : n ∈ ℕ )

( (0.5 − 1/n, 0.5 + 1/n) : n ∈ ℕ )

( (0, 1/n) : n ∈ ℕ )

These intervals shrink inward as n→∞ and no finite subcollection covers near 0 or 1.

Explanation

These intervals shrink inward as n→∞ and no finite subcollection covers near 0 or 1.

2) If there exists even one open cover of a space X that has no finite subcover, then:

X is compact

X is Lindelöf

X is not compact

X must be bounded

One failure means the space is not compact.

Explanation

One failure means the space is not compact.

3) Which statement is true about closed subsets of compact spaces?

Always open

They are compact

They are bounded

They are finite

Closed subsets inherit compactness.

Explanation

Closed subsets inherit compactness.

4) Which statement is true about continuous images of compact sets?

Always open

Always closed

They are compact

Bounded only if domain is bounded

Continuous images of compact sets remain compact.

Explanation

Continuous images of compact sets remain compact.

5) In ℝ, which of the following is necessarily non-compact?

[-1,1]

[0,1/2]

[0,∞)

{0}

[0,∞) is unbounded.

Explanation

[0,∞) is unbounded.

6) In ℝ, an open cover of (0,1) can fail to have a finite subcover.

True

False

(0,1) is not compact.

Explanation

(0,1) is not compact.

7) Every finite subcover of a compact set’s open cover must consist of open sets.

True

False

Subcovers consist of sets from the original open cover.

Explanation

Subcovers consist of sets from the original open cover.

8) The existence of a finite subcover for one open cover is enough to conclude that a space is compact.

True

False

All open covers must have finite subcovers.

Explanation

All open covers must have finite subcovers.

9) If a set is compact, then removing a single point preserves compactness.

True

False

Removing finitely many points preserves compactness.

Explanation

Removing finitely many points preserves compactness.

10) If an open cover has no finite subcover, any larger open cover also has no finite subcover.

True

False

Adding sets may fix the issue.

Explanation

Adding sets may fix the issue.

11) In ℝⁿ, a set is compact iff closed and bounded.

True

False

Heine–Borel theorem.

Explanation

Heine–Borel theorem.

12) Every bounded subset of ℝ is compact.

True

False

Must also be closed.

Explanation

Must also be closed.

13) Every infinite open cover of a compact space can be reduced to a finite subcover.

True

False

This is the defining property.

Explanation

This is the defining property.

14) If one open cover fails to admit a finite subcover, the space is not compact.

True

False

Compactness requires all open covers to have finite subcovers.

Explanation

Compactness requires all open covers to have finite subcovers.

15) The union of finitely many compact subsets is compact.

True

False

Finite unions preserve compactness.

Explanation

Finite unions preserve compactness.

Finite Subcovers Application Quiz

1) Which family is an open cover of (0,1) that has no finite subcover?

2)

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

2) If there exists even one open cover of a space X that has no finite subcover, then:

3) Which statement is true about closed subsets of compact spaces?

4) Which statement is true about continuous images of compact sets?

5) In ℝ, which of the following is necessarily non-compact?

6) In ℝ, an open cover of (0,1) can fail to have a finite subcover.

7) Every finite subcover of a compact set’s open cover must consist of open sets.

8) The existence of a finite subcover for one open cover is enough to conclude that a space is compact.

9) If a set is compact, then removing a single point preserves compactness.

10) If an open cover has no finite subcover, any larger open cover also has no finite subcover.

11) In ℝⁿ, a set is compact iff closed and bounded.

12) Every bounded subset of ℝ is compact.

13) Every infinite open cover of a compact space can be reduced to a finite subcover.

14) If one open cover fails to admit a finite subcover, the space is not compact.

15) The union of finitely many compact subsets is compact.