Finite Subcovers Quiz

1) An open cover of a set X is a collection of open sets whose union contains X.

True

False

This is the definition of an open cover: many open sets whose union includes the whole set.

Explanation

This is the definition of an open cover: many open sets whose union includes the whole set.

2) A finite subcover is any finite subcollection of a cover.

True

False

A finite subcover must still cover X. A random finite subcollection might not.

Explanation

A finite subcover must still cover X. A random finite subcollection might not.

3) A topological space is compact if and only if every open cover has a finite subcover.

True

False

This is the formal definition of compactness.

Explanation

This is the formal definition of compactness.

4) If there exists at least one open cover of a space that has a finite subcover, then the space is compact.

True

False

Compactness requires every open cover to have a finite subcover, not just one.

Explanation

Compactness requires every open cover to have a finite subcover, not just one.

5) The closed interval [0,1] ⊆ ℝ is compact because every open cover of it has a finite subcover.

True

False

By the Heine–Borel theorem, closed and bounded intervals in ℝ are compact.

Explanation

By the Heine–Borel theorem, closed and bounded intervals in ℝ are compact.

6) The open interval (0,1) ⊆ ℝ is not compact because there exists an open cover with no finite subcover.

True

False

Example cover: {(1/n,1) : n∈N} has no finite subcover.

Explanation

Example cover: {(1/n,1) : n∈N} has no finite subcover.

7) Every closed subset of a compact space is compact.

True

False

Closed subsets inherit compactness.

Explanation

Closed subsets inherit compactness.

8) Every open subset of a compact space is compact.

True

False

Open subsets may fail to be compact.

Explanation

Open subsets may fail to be compact.

9) The continuous image of a compact space is compact.

True

False

Continuity preserves compactness under images.

Explanation

Continuity preserves compactness under images.

10) In any Hausdorff space, compact subsets are closed.

True

False

Compact sets in Hausdorff spaces are closed.

Explanation

Compact sets in Hausdorff spaces are closed.

11) Which best defines an open cover of a set X in a topological space?

A collection of sets whose intersections equal X

A finite collection of open sets whose union equals X

A collection of open sets whose union contains X

A single open set that equals X

Union must contain X; not necessarily equal or finite.

Explanation

Union must contain X; not necessarily equal or finite.

12) What is a finite subcover?

Any finite subset of a cover

A finite subcollection that still covers the entire set

A finite subcollection of closed sets

A finite subcollection whose intersections cover the set

A subcover must continue to cover the whole set.

Explanation

A subcover must continue to cover the whole set.

13) A topological space is compact if and only if:

Every closed cover has a finite subcover

Every open cover has a finite subcover

There exists at least one open cover with a finite subcover

It is bounded

Compactness is defined using open covers.

Explanation

Compactness is defined using open covers.

14) In ℝⁿ with the usual topology, which condition characterizes compact sets?

Closed only

Bounded only

Closed and bounded

Open and bounded

Heine–Borel theorem.

Explanation

Heine–Borel theorem.

15) Which of the following subsets of ℝ is compact?

(0,1)

[0,1]

[0,∞)

ℝ

[0,1] is closed and bounded.

Explanation

[0,1] is closed and bounded.

Finite Subcovers Quiz

1) An open cover of a set X is a collection of open sets whose union contains X.

2)

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

2) A finite subcover is any finite subcollection of a cover.

3) A topological space is compact if and only if every open cover has a finite subcover.

4) If there exists at least one open cover of a space that has a finite subcover, then the space is compact.

5) The closed interval [0,1] ⊆ ℝ is compact because every open cover of it has a finite subcover.

6) The open interval (0,1) ⊆ ℝ is not compact because there exists an open cover with no finite subcover.

7) Every closed subset of a compact space is compact.

8) Every open subset of a compact space is compact.

9) The continuous image of a compact space is compact.

10) In any Hausdorff space, compact subsets are closed.

11) Which best defines an open cover of a set X in a topological space?

12) What is a finite subcover?

13) A topological space is compact if and only if:

14) In ℝⁿ with the usual topology, which condition characterizes compact sets?

15) Which of the following subsets of ℝ is compact?