Identifying Interior Points Quiz

1) Let A = (0,5). Which are interior points?

0

1

4.9

5

Interior points of (0,5) are all numbers strictly between 0 and 5.

Explanation

Interior points of (0,5) are all numbers strictly between 0 and 5.

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2) Interior of (0,2) ∪ {5}:

0.5

1

2

5

(0,2) is open, so points between 0 and 2 are interior. The point 5 is isolated, not interior.

Explanation

(0,2) is open, so points between 0 and 2 are interior. The point 5 is isolated, not interior.

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3) Let A = [0,4] \ [1,2]. Interior points:

0.1

1.5

3

4

The set is [0,1) ∪ (2,4]. The interior is (0,1) ∪ (2,4). So 0.1 and 3 are interior; 1.5 is removed, and 4 is an endpoint.

Explanation

The set is [0,1) ∪ (2,4]. The interior is (0,1) ∪ (2,4). So 0.1 and 3 are interior; 1.5 is removed, and 4 is an endpoint.

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4) Which sets contain at least one interior point?

(0,1)

{5}

(−3,−1)

ℤ

Open intervals have interior points. Singletons and ℤ (discrete points) have none.

Explanation

Open intervals have interior points. Singletons and ℤ (discrete points) have none.

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5) Which statements about interior points are correct?

Every interior point belongs to the set

Interior points form an open set

The interior may be empty

Interior points must be boundary points

Interior points lie inside the set, form an open set, and some sets have empty interior.

Explanation

Interior points lie inside the set, form an open set, and some sets have empty interior.

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6) A point x is not an interior point of a set A when:

No open interval around x fits inside A

Every open interval intersects A

X is an endpoint

X is isolated

If no open interval stays inside the set, x is not an interior point.

Explanation

If no open interval stays inside the set, x is not an interior point.

7) Interior of ℝ \ {0}:

All real numbers except 0

Includes 0

(−∞,0) ∪ (0,∞)

Empty

Removing one point leaves an open set: ℝ minus 0.

Explanation

Removing one point leaves an open set: ℝ minus 0.

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8) A point fails to be an interior point:

Every open interval around it contains points outside A

It is in the interior of A

It is a boundary point

It has no neighborhood contained in A

Interior requires a full interval inside the set. If none exists, the point is not interior.

Explanation

Interior requires a full interval inside the set. If none exists, the point is not interior.

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9) Int(ℝ \ (2,7)) includes:

1

2

10

7

ℝ \ (2,7) = (−∞,2] ∪ [7,∞). Interior is (−∞,2) ∪ (7,∞). So 1 and 10 are interior points; 2 and 7 are boundary points.

Explanation

ℝ \ (2,7) = (−∞,2] ∪ [7,∞). Interior is (−∞,2) ∪ (7,∞). So 1 and 10 are interior points; 2 and 7 are boundary points.

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10) Int(ℝ \ ℤ) includes:

1.5

1

3.7

3

ℝ minus integers is open everywhere except the integers themselves.

Explanation

ℝ minus integers is open everywhere except the integers themselves.

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11) Int(ℝ \ {1/n : n ∈ ℕ}) includes:

0

0.5

1

Every real number except countably many isolated points

Removing isolated points leaves an open set everywhere except those points.

Explanation

Removing isolated points leaves an open set everywhere except those points.

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12) Set A = ∪ₙ₌₁∞ (1/n, 1 − 1/n). Interior points include:

0.5

0.1

0.9

1

As n grows, the intervals approach (0,1). Interior becomes (0,1).

Explanation

As n grows, the intervals approach (0,1). Interior becomes (0,1).

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13) Set A = ∩ₙ₌₁∞ (−1/n, 2 + 1/n). Interior points include:

0

1

−0.5

2

Intersection of these shrinking intervals is [0,2]. Interior is (0,2).

Explanation

Intersection of these shrinking intervals is [0,2]. Interior is (0,2).

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14) Let A = (0,1) ∪ (1,2) ∪ {1/2}. Interior points include:

0.25

0.5

1

1.75

Interior is (0,1) ∪ (1,2). 0.5 is an isolated point here.

Explanation

Interior is (0,1) ∪ (1,2). 0.5 is an isolated point here.

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15) Let A = (0,1) ∪ (1,2) ∪ {2} ∪ (3,4). Interior points include:

0.2

1

3.5

4

Interior is (0,1), (1,2), and (3,4). The point 2 and 4 are not interior.

Explanation

Interior is (0,1), (1,2), and (3,4). The point 2 and 4 are not interior.

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Identifying Interior Points Quiz

1) Let A = (0,5). Which are interior points?

2)

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

2) Interior of (0,2) ∪ {5}:

3) Let A = [0,4] \ [1,2]. Interior points:

4) Which sets contain at least one interior point?

5) Which statements about interior points are correct?

6) A point x is not an interior point of a set A when:

7) Interior of ℝ \ {0}:

8) A point fails to be an interior point:

9) Int(ℝ \ (2,7)) includes:

10) Int(ℝ \ ℤ) includes:

11) Int(ℝ \ {1/n : n ∈ ℕ}) includes:

12) Set A = ∪ₙ₌₁∞ (1/n, 1 − 1/n). Interior points include:

13) Set A = ∩ₙ₌₁∞ (−1/n, 2 + 1/n). Interior points include:

14) Let A = (0,1) ∪ (1,2) ∪ {1/2}. Interior points include:

15) Let A = (0,1) ∪ (1,2) ∪ {2} ∪ (3,4). Interior points include: