Closure of Sets Properties Quiz

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| By Thames
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Thames
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Quizzes Created: 8157 | Total Attempts: 9,566,492
| Questions: 15 | Updated: Dec 12, 2025
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Question 1 / 16
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1) The closure of the intersection of two sets equals the intersection of their closures.

Explanation

Generally, (A ∩ B)̄ ⊆ Ā ∩ B̄, but not equal.

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About This Quiz
Closure Of Sets Properties Quiz - Quiz

Think you understand how closures behave? This quiz helps you explore the deeper properties that define closures in topology. You’ll analyze how closures interact with unions, intersections, complements, and limit points. You’ll also identify closures of common sets, understand when closure equals the set itself, and determine how closure connects... see moreto boundary behavior. These examples help you develop a clearer picture of how closures complete sets and why they play a central role in mathematical analysis. By the end, you’ll be ready to apply closure rules confidently across a variety of problems!
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2)
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2) The closure of the complement of a set equals the complement of its interior.

Explanation

Always true: closure(Aᶜ) = (interior(A))ᶜ.

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3) If set A has a limit point x, then x is in A.

Explanation

Limit points need not lie in A (example: A=(0,1), limit point=0).

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4) The closure of an infinite set always contains infinitely many points.

Explanation

An infinite set’s closure cannot collapse to a finite set.

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5) In ℝ, the closure of (0,1) ∪ (1,2) is [0,2].

Explanation

The endpoints 0, 1, and 2 are all limit points, so closure = [0,2].

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6) In ℝ², the closure of the set ({(x, 1/x) : x > 0}) contains the point (0,∞).

Explanation

No finite point approaches (0,∞); the graph has no real limit point at x = 0.

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7) Which of the following best defines a closure of a set A?

Explanation

The closure is the smallest closed set that contains A.

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8) The closure of a set always includes:

Explanation

The closure contains A plus its limit points, so it always contains A itself.

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9) The closure of the empty set ∅ is:

Explanation

The empty set has no limit points, so its closure is ∅.

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10) The closure of an open set is:

Explanation

Closures are always closed sets.

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11) The closure of (0,1) ∪ {5} is:

Explanation

The closure includes 0 and 1 (endpoints) and the isolated point 5.

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12) If A ⊆ B, then:

Explanation

If A is a subset of B, then A ⊆ B is the correct statement.

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13) Which of the following is always true?

Explanation

A set union equals itself; this is always true.

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14) A set with no limit points has its closure equal to the set itself.

Explanation

Such a set contains all its (nonexistent) limit points, so the closure adds nothing.

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15) The closure of the union of two sets equals the union of their closures.

Explanation

Closure distributes over unions: (Ā ∪ B̄).

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The closure of the intersection of two sets equals the intersection of...
The closure of the complement of a set equals the complement of its...
If set A has a limit point x, then x is in A.
The closure of an infinite set always contains infinitely many points.
In ℝ, the closure of (0,1) ∪ (1,2) is [0,2].
In ℝ², the closure of the set ({(x, 1/x) : x > 0}) contains the...
Which of the following best defines a closure of a set A?
The closure of a set always includes:
The closure of the empty set ∅ is:
The closure of an open set is:
The closure of (0,1) ∪ {5} is:
If A ⊆ B, then:
Which of the following is always true?
A set with no limit points has its closure equal to the set itself.
The closure of the union of two sets equals the union of their...
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