Boundary Points Quiz

1) A boundary point of a set is a point where every open interval around it contains points from the set.

True

False

Boundary points require every neighborhood to meet the set.

Explanation

Boundary points require every neighborhood to meet the set.

2) Boundary points must always lie inside the set.

True

False

They can be inside or outside.

Explanation

They can be inside or outside.

3) Every open interval has no boundary points.

True

False

An open interval has boundary points—its endpoints.

Explanation

An open interval has boundary points—its endpoints.

4) If A is closed, then all of its points are boundary points.

True

False

Closed intervals have interior points too.

Explanation

Closed intervals have interior points too.

5) The boundary of a closed set is always closed.

True

False

Boundaries are always closed.

Explanation

Boundaries are always closed.

6) The boundary of a closed interval [a,b] is the set {a,b}.

True

False

The endpoints are exactly its boundary.

Explanation

The endpoints are exactly its boundary.

7) The empty set has an empty boundary.

True

False

No points = no boundary points.

Explanation

No points = no boundary points.

8) If A is open, then:

It has no boundary points

The boundary is ℝ

The boundary may lie outside A

The boundary is A

Open sets can have boundaries outside them.

Explanation

Open sets can have boundaries outside them.

9) If A is closed, which is true?

All points are boundary points

Boundary points ⊆ A

A has no boundary points

A = ℝ

Closed sets contain all their boundary points.

Explanation

Closed sets contain all their boundary points.

10) The boundary of ℚ is:

∅

ℚ

ℝ

ℚᶜ

Rationals are dense; boundary is all real numbers.

Explanation

Rationals are dense; boundary is all real numbers.

11) The boundary of a union of disjoint intervals can be:

Finite

Infinite

Either A or B

Always infinite

Depends on how many disjoint intervals there are.

Explanation

Depends on how many disjoint intervals there are.

12) If cl(A) = int(A), then boundary(A) is:

∅

A

ℝ

Aᶜ

Boundary = closure – interior = ∅.

Explanation

Boundary = closure – interior = ∅.

13) A point x is a boundary point of A iff:

X ∉ cl(A)

X ∉ cl(A) ∩ cl(Aᶜ)

X ∉ ℝ

X ∈ int(A)

Boundary = cl(A) ∩ cl(Aᶜ).

Explanation

Boundary = cl(A) ∩ cl(Aᶜ).

14) If A has an empty boundary, which must be true?

A is finite

A is either ∅ or ℝ

A has no isolated points

A is countable

Only ∅ and ℝ have no boundary.

Explanation

Only ∅ and ℝ have no boundary.

15) The boundary of A ∪ B equals boundary(A) ∪ boundary(B) only if:

A and B overlap

A and B are disjoint and sufficiently separated

A and B share the same closure

A and B are finite

Works only when boundaries do not create new boundary points.

Explanation

Works only when boundaries do not create new boundary points.

Boundary Points Quiz

1) A boundary point of a set is a point where every open interval around it contains points from the set.

2)

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

2) Boundary points must always lie inside the set.

3) Every open interval has no boundary points.

4) If A is closed, then all of its points are boundary points.

5) The boundary of a closed set is always closed.

6) The boundary of a closed interval [a,b] is the set {a,b}.

7) The empty set has an empty boundary.

8) If A is open, then:

9) If A is closed, which is true?

10) The boundary of ℚ is:

11) The boundary of a union of disjoint intervals can be:

12) If cl(A) = int(A), then boundary(A) is:

13) A point x is a boundary point of A iff:

14) If A has an empty boundary, which must be true?

15) The boundary of A ∪ B equals boundary(A) ∪ boundary(B) only if: