Compactness And Finite Subcovers Mastery Quiz

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

True

False

This is the definition of an open cover.

Explanation

This is the definition of an open cover.

2) A finite subcover must consist of all the sets in the original cover.

True

False

A finite subcover only requires finitely many sets, not all of them.

Explanation

A finite subcover only requires finitely many sets, not all of them.

3) If even one open cover of a space has no finite subcover, the space is not compact.

True

False

Compactness requires every open cover to admit a finite subcover.

Explanation

Compactness requires every open cover to admit a finite subcover.

4) A finite subcover always exists for any open cover of any topological space.

True

False

Non-compact spaces have open covers with no finite subcover.

Explanation

Non-compact spaces have open covers with no finite subcover.

5) If a space is compact, every open cover of the space has at least one finite subcover.

True

False

This is the definition of compactness.

Explanation

This is the definition of compactness.

6) Being bounded is enough to guarantee finite subcovers exist for all open covers.

True

False

Bounded sets like (0,1) are not compact.

Explanation

Bounded sets like (0,1) are not compact.

7) If a cover already consists of finitely many sets, it automatically contains a finite subcover.

True

False

A finite cover is itself a finite subcover.

Explanation

A finite cover is itself a finite subcover.

8) A finite subcover must be a minimal collection of sets covering the space.

True

False

Finite does not mean minimal.

Explanation

Finite does not mean minimal.

9) Compactness of a subset depends entirely on whether finite subcovers exist for open covers.

True

False

Compactness is defined exactly this way.

Explanation

Compactness is defined exactly this way.

10) If every open cover of a set has a finite subcover, then the set must also be closed.

True

False

Compact subsets need not be closed in non-Hausdorff spaces.

Explanation

Compact subsets need not be closed in non-Hausdorff spaces.

11) Which of the following statements about (0,1] is true?

It is compact because it is bounded

It is compact because it is closed in R

It is not compact; e.g. ((1/n,2)) is an open cover with no finite subcover

It is compact because every open cover has a countable subcover

(0,1] is not closed in R and not compact.

Explanation

(0,1] is not closed in R and not compact.

12) Which statement correctly completes: “Every open cover of a compact space…”

Has a countable subcover

Has a finite subcover

Is finite

Is disjoint

Finite subcover = compactness.

Explanation

Finite subcover = compactness.

13) Which is guaranteed for a continuous function f:K→R where K is compact?

F is Lipschitz

F is uniformly continuous

F is differentiable

F is injective

Continuous functions on compact sets are uniformly continuous.

Explanation

Continuous functions on compact sets are uniformly continuous.

14) Extreme Value Theorem follows from compactness. Which is true?

If K is compact and f is continuous, f attains max/min

If K is bounded and f is continuous, f attains max

If f is continuous, it attains extrema on any domain

If K is closed, f attains a max on K

Compactness ensures extrema.

Explanation

Compactness ensures extrema.

15) Which statement about unions of compact sets is true?

Any union of compact sets is compact

Countable unions of compact sets are always compact

Finite unions of compact sets are compact

Intersections of compact sets are never compact

Finite unions remain compact.

Explanation

Finite unions remain compact.

Compactness and Finite Subcovers Mastery Quiz

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

2)

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

2) A finite subcover must consist of all the sets in the original cover.

3) If even one open cover of a space has no finite subcover, the space is not compact.

4) A finite subcover always exists for any open cover of any topological space.

5) If a space is compact, every open cover of the space has at least one finite subcover.

6) Being bounded is enough to guarantee finite subcovers exist for all open covers.

7) If a cover already consists of finitely many sets, it automatically contains a finite subcover.

8) A finite subcover must be a minimal collection of sets covering the space.

9) Compactness of a subset depends entirely on whether finite subcovers exist for open covers.

10) If every open cover of a set has a finite subcover, then the set must also be closed.

11) Which of the following statements about (0,1] is true?

12) Which statement correctly completes: “Every open cover of a compact space…”

13) Which is guaranteed for a continuous function f:K→R where K is compact?

14) Extreme Value Theorem follows from compactness. Which is true?

15) Which statement about unions of compact sets is true?