Identifying Interior Points Quiz

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Quizzes Created: 7288 | Total Attempts: 9,526,295
| Questions: 15 | Updated: Nov 24, 2025
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1) Let A = (0,5). Which are interior points?

Explanation

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

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About This Quiz
Identifying Interior Points Quiz - Quiz

How well can you identify interior points by looking at the structure of a set? This quiz lets you put your knowledge into action. You’ll analyze intervals, unions, complements, and removed points to determine which values belong to the interior. Using open intervals and neighborhood definitions, you'll test your ability... see moreto spot points that lie fully within a set. With examples ranging from simple intervals to infinite unions and complements of integers, you’ll gain confidence in recognizing interior points in a variety of mathematical situations! see less

2)
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2) Interior of (0,2) ∪ {5}:

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:

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?

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?

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:

Explanation

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

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7) Interior of ℝ \ {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:

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:

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:

Explanation

ℝ minus integers is open everywhere except the integers themselves.

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11) Int(ℝ \ {1/n : n ∈ ℕ}) includes:

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:

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:

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:

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:

Explanation

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

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Let A = (0,5). Which are interior points?
Interior of (0,2) ∪ {5}:
Let A = [0,4] \ [1,2]. Interior points:
Which sets contain at least one interior point?
Which statements about interior points are correct?
A point x is not an interior point of a set A when:
Interior of ℝ \ {0}:
A point fails to be an interior point:
Int(ℝ \ (2,7)) includes:
Int(ℝ \ ℤ) includes:
Int(ℝ \ {1/n : n ∈ ℕ}) includes:
Set A = ∪ₙ₌₁∞ (1/n, 1 − 1/n). Interior points...
Set A = ∩ₙ₌₁∞ (−1/n, 2 + 1/n). Interior points...
Let A = (0,1) ∪ (1,2) ∪ {1/2}. Interior points include:
Let A = (0,1) ∪ (1,2) ∪ {2} ∪ (3,4). Interior points...
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