Closure of Sets 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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1) If every open interval around a point x contains at least one point of a set A, then x must be in the closure of A.

Explanation

This is the definition of closure: every neighborhood touches the set.

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About This Quiz
Closure Of Sets Quiz - Quiz

Ready to learn what it means for a set to be “closed off” in topology? This quiz helps you understand the closure of a set by exploring limit points, accumulation behavior, and the relationship between open and closed sets. You’ll examine closures of intervals, unions, discrete sets, and classic examples... see morelike ℤ and ℚ in ℝ. Through these practice questions, you’ll build a strong intuition for how closures extend sets and why they matter in topological structure. By the end, you’ll confidently determine closures and understand how mathematicians describe the boundaries of sets!
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2) The closure of a set A is always a closed set.

Explanation

By definition, the closure is the smallest closed set containing A.

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3) The closure of an empty set is the empty set.

Explanation

∅ has no limit points, so its closure is ∅.

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4) The closure of a single point set {a} is {a}.

Explanation

A singleton has no other limit points in ℝ, so its closure is itself.

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5) If a set A is closed then A = Ā.

Explanation

Closed sets already contain all their limit points.

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6) If a set A is already closed, then its closure is equal to A.

Explanation

The closure cannot add new points to a closed set.

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7) If a set A is open, then its closure is always open.

Explanation

Example: (0,1) is open, but its closure [0,1] is closed, not open.

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8) Which is generally false?

Explanation

As written, A ∩ B ⊆ A ∩ B conveys no new information; the intended meaning is false.

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9) A point x is not in the closure of A if and only if:

Explanation

If one neighborhood misses A completely, x is not in the closure.

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10) The closure of (0,1) ∪ (1,2):

Explanation

(0,1) ∪ (1,2) accumulates to 0, 1, and 2. Closure = [0,2].

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11) Which identity is always true?

Explanation

Complement of A equals complement of its interior’s closure relation: int(A)ᶜ = closure(Aᶜ).

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12) In ℝ, the closure of ℤ is:

Explanation

Integers have no limit points besides themselves in ℝ.

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13) A set can have no interior but still have nonempty closure. Which example shows this?

Explanation

ℚ has empty interior but dense closure (its closure is ℝ).

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14) In ℝ², the closure of the graph of y = 1/x for x > 0 is:

Explanation

As x > 0, the graph has no additional finite accumulation points—its closure is the graph itself.

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15) A set with all its limit points removed has closure equal to:

Explanation

Removing limit points leaves only isolated points, whose closure is the isolated points themselves.

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If every open interval around a point x contains at least one point of...
The closure of a set A is always a closed set.
The closure of an empty set is the empty set.
The closure of a single point set {a} is {a}.
If a set A is closed then A = Ā.
If a set A is already closed, then its closure is equal to A.
If a set A is open, then its closure is always open.
Which is generally false?
A point x is not in the closure of A if and only if:
The closure of (0,1) ∪ (1,2):
Which identity is always true?
In ℝ, the closure of ℤ is:
A set can have no interior but still have nonempty closure. Which...
In ℝ², the closure of the graph of y = 1/x for x > 0 is:
A set with all its limit points removed has closure equal to:
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