Closure Of Sets Quiz

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.

True

False

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

Explanation

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

2) The closure of a set A is always a closed set.

True

False

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

Explanation

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

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

True

False

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

Explanation

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

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

True

False

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

Explanation

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

5) If a set A is closed then A = Ā.

True

False

Closed sets already contain all their limit points.

Explanation

Closed sets already contain all their limit points.

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

True

False

The closure cannot add new points to a closed set.

Explanation

The closure cannot add new points to a closed set.

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

True

False

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

Explanation

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

8) Which is generally false?

A ∩ B ⊆ A ∩ B

A ∩ B = A ∩ B

A ∪ B = A ∪ B

Limit points belong to closures

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

Explanation

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

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

Every open set around x intersects A

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

A is closed

X = 0

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

Explanation

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

10) The closure of (0,1) ∪ (1,2):

[0,2]

[0,1) ∪ (1,2]

[0,1] ∪ [1,2]

∅

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

Explanation

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

11) Which identity is always true?

Aᶜ = int(A)ᶜ

Aᶜ = A

Aᶜ = Ā

Aᶜ = int(A)

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

Explanation

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

12) In ℝ, the closure of ℤ is:

∅

ℤ

ℝ

(0,1)

Integers have no limit points besides themselves in ℝ.

Explanation

Integers have no limit points besides themselves in ℝ.

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

∅

(0,1)

ℚ

[0,1]

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

Explanation

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

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

Entire plane

Entire x-axis

The graph plus the vertical accumulation at infinity

Only the graph itself

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

Explanation

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

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

The empty set

Its set of isolated points

The whole space

Undefined

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

Explanation

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

Closure of Sets Quiz

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.

2)

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

2) The closure of a set A is always a closed set.

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

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

5) If a set A is closed then A = Ā.

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

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

8) Which is generally false?

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

10) The closure of (0,1) ∪ (1,2):

11) Which identity is always true?

12) In ℝ, the closure of ℤ is:

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

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

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