Identifying Closures Quiz

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1) Intersection of closed supersets

Union of all open subsets

All points where every neighborhood intersects the set

Set of all limit points + original set

Closure can be defined as: intersection of all closed sets containing A, points whose every neighborhood intersects A, A together with all its limit points.

Explanation

Closure can be defined as: intersection of all closed sets containing A, points whose every neighborhood intersects A, A together with all its limit points.

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About This Quiz

How well can you determine the closure of a set in a topological space? This quiz lets you put your knowledge into action. You’ll identify closures using limit points, closed supersets, boundary behavior, and neighborhood intersections. Through examples using discrete sets, convergent sequences, complements, and infinite unions, you’ll get hands-on... see morepractice with how closures extend sets and capture their limit behavior. By the end, you’ll be able to compute and recognize closures quickly, whether in ℝ or in general topological spaces!
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2)

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2) Cl(A)

∂A

cl(A) is standard notation for closure.

Explanation

cl(A) is standard notation for closure.

3) Every open set around x intersects A

Some open set around x intersects A

X is a limit point

X ∈ A

A is in the closure if it is in A or is a limit point (i.e., all neighborhoods intersect A).

Explanation

A is in the closure if it is in A or is a limit point (i.e., all neighborhoods intersect A).

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4) It contains all points of A

It contains all limit points of A

It is always smaller than A

It is the smallest closed set containing A

Closure = A plus all its limit points, forming the smallest closed superset.

Explanation

Closure = A plus all its limit points, forming the smallest closed superset.

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5) Isolated points in A

Limit points of A

Boundary points of A

Exterior points of A

Closure contains A, all limit points, and all boundary points.

Explanation

Closure contains A, all limit points, and all boundary points.

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6) [0,1]

∅

ℝ

(0,1)

Closed sets include their boundary points: [0,1], ∅, and ℝ.

Explanation

Closed sets include their boundary points: [0,1], ∅, and ℝ.

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7) (0,1)

(0,1] ∪ {0.2}

{1/n : n ∈ ℕ} ∪ {1}

{1/n}

(0,1) has closure [0,1]; the others do not include all points of [0,1].

Explanation

(0,1) has closure [0,1]; the others do not include all points of [0,1].

8) Equal to A

Larger than A

Empty

Smaller than A

The interior can equal A (if A is open), be empty, or be smaller—but never larger.

Explanation

The interior can equal A (if A is open), be empty, or be smaller—but never larger.

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9) There exists an open set around x that does not intersect A

X ∉ A

X is isolated from A

X is a boundary point of A

x is not in the closure if a neighborhood avoids A completely.

Explanation

x is not in the closure if a neighborhood avoids A completely.

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10) Contains a single limit point

Contains infinitely many isolated points

Contains 0

Contains 1/2

Sequence 1/n converges to 0, so closure = {1/n} ∪ {0}.

Explanation

Sequence 1/n converges to 0, so closure = {1/n} ∪ {0}.

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11) A ∩ B = Ā ∩ B̄ always

∅̄ = ∅

X̄ = X

A ∪ B = Ā ∪ B̄

Closure of ∅ is ∅, closure of X is X, and closure distributes over unions.

Explanation

Closure of ∅ is ∅, closure of X is X, and closure distributes over unions.

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12) 0

Isolated points only

1/n → 0 and (2 − 1/n) → 2, so closure contains 0 and 2.

Explanation

1/n → 0 and (2 − 1/n) → 2, so closure contains 0 and 2.

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13) Extensive: A ⊆ Ā

Monotone: A ⊆ B ⇒ Ā ⊆ B̄

Idempotent

Complement preserving

Closure must include the set, preserve subset relations, and satisfy Ā̄ = Ā.

Explanation

Closure must include the set, preserve subset relations, and satisfy Ā̄ = Ā.

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14) Contains 0

Contains 1/10

Contains no integer

Contains values > 1

1/n still converges to 0, and 1/10 is in the set.

Explanation

1/n still converges to 0, and 1/10 is in the set.

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15) 0

Any number > 1

Limit point at 0

The values get arbitrarily small, so closure includes limit point 0.

Explanation

The values get arbitrarily small, so closure includes limit point 0.

Thank you for your feedback!

Jede Crisle Cortes Davila |Bachelor of Engineering |

College Expert

Jede Crisle D. is a mathematics subject matter expert specializing in Algebra, Geometry, and Calculus. She focuses on developing clear, solution-driven mathematical explanations and has strong experience with LaTeX-based math content. She holds a Bachelor’s degree in Electronics and Communications Engineering.