Interior Points Quiz

1) An interior of a set U is a point for which there exists an open ball completely contained in U.

True

False

This is the definition of an interior point—there must be a small open ball inside the set.

Explanation

This is the definition of an interior point—there must be a small open ball inside the set.

2) Every interior of a set U must also be a boundary point of that set.

True

False

Interior points are inside the set, boundary points are on the 'edge.' An interior point is not a boundary point.

Explanation

Interior points are inside the set, boundary points are on the 'edge.' An interior point is not a boundary point.

3) The interior of a set U is always an open set.

True

False

The interior is defined as the largest open set inside U, so it is always open.

Explanation

The interior is defined as the largest open set inside U, so it is always open.

4) If set U is open, then every point in the set is an interior point.

True

False

In an open set, every point has a small open interval around it inside the set.

Explanation

In an open set, every point has a small open interval around it inside the set.

5) If set U is closed, then it cannot contain any interior points.

True

False

A set can be closed and still have interior points—for example, the interval [0,1] has interior (0,1).

Explanation

A set can be closed and still have interior points—for example, the interval [0,1] has interior (0,1).

6) The empty set ∅ has no interior points.

True

False

There are no points in ∅, so it has no interior points.

Explanation

There are no points in ∅, so it has no interior points.

7) A point can be an interior point of U even if it lies outside of U.

True

False

Interior points must lie inside the set.

Explanation

Interior points must lie inside the set.

8) Which statement is true?

Int(A ∪ B) = int(A) ∪ int(B)

Int(A ∪ B) always contains int(A) ∪ int(B)

Int(A ∪ B) equals A ∪ B

Int(A ∪ B) = ∅

The interior of a union is at least as big as the union of interiors.

Explanation

The interior of a union is at least as big as the union of interiors.

9) If A is closed and has a nonempty interior, then:

It cannot be bounded

It must be equal to ℝ

It contains only isolated points

It contains an open interval

Having a nonempty interior means A includes some open interval.

Explanation

Having a nonempty interior means A includes some open interval.

10) Which set has a nonempty interior?

{1/n : n ∈ ℕ}

ℤ

(0,1) ∪ {5}

Cantor set

(0,1) is an open interval, so it has interior points.

Explanation

(0,1) is an open interval, so it has interior points.

11) If x is an interior point of A and A ⊆ B, then:

X is also an interior point of B

X may not be in B

X is not in the closure of B

B must be open

If a ball around x is inside A, it is also inside B, so x is interior to B.

Explanation

If a ball around x is inside A, it is also inside B, so x is interior to B.

12) Which statement is true?

Int(A ∩ B) = int(A) ∪ int(B)

Int(A ∩ B) = int(A) ∩ int(B)

Int(A ∩ B) = A ∩ B

Int(A ∩ B) = ∅

The interior of an intersection is the intersection of the interiors.

Explanation

The interior of an intersection is the intersection of the interiors.

13) Suppose A is open and int(B) ⊆ A, then:

B must be open

A must be dense

B contains an open subset inside A

A ⊆ B

If int(B) ⊆ A, then B has at least some open part that lies inside A.

Explanation

If int(B) ⊆ A, then B has at least some open part that lies inside A.

14) The statement int(A) ⊆ A is:

Always true

True only for open sets

True only for closed sets

Always false

The interior of a set is always made of points inside that set.

Explanation

The interior of a set is always made of points inside that set.

15) A point is not an interior point if:

No open ball around it lies fully in the set

It is in an open interval

It lies in the interior

The set is open

If no open ball stays inside the set, then the point is not interior.

Explanation

If no open ball stays inside the set, then the point is not interior.

Interior Points Quiz

1) An interior of a set U is a point for which there exists an open ball completely contained in U.

2)

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

2) Every interior of a set U must also be a boundary point of that set.

3) The interior of a set U is always an open set.

4) If set U is open, then every point in the set is an interior point.

5) If set U is closed, then it cannot contain any interior points.

6) The empty set ∅ has no interior points.

7) A point can be an interior point of U even if it lies outside of U.

8) Which statement is true?

9) If A is closed and has a nonempty interior, then:

10) Which set has a nonempty interior?

11) If x is an interior point of A and A ⊆ B, then:

12) Which statement is true?

13) Suppose A is open and int(B) ⊆ A, then:

14) The statement int(A) ⊆ A is:

15) A point is not an interior point if: