Identifying Boundary Points Quiz

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1) Which of the following are true about a boundary point of set A?

Every neighborhood of the point intersects A

Every neighborhood of the point intersects Aᶜ

The point must be in A

The point may be outside A

A boundary point must have every neighborhood touching both the set A and its complement Aᶜ, so (A) and (B) are true. It can lie inside or outside the set, so (D) is true.

Explanation

A boundary point must have every neighborhood touching both the set A and its complement Aᶜ, so (A) and (B) are true. It can lie inside or outside the set, so (D) is true.

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About This Quiz

How well can you spot the boundary of a set? This quiz puts your understanding into action by exploring sets with different shapes and structures. You’ll work with intervals, complements, discrete sets, unions, and classic fractal examples to determine where boundary points lie. Using the definitions involving closure and interior,... see moreyou’ll test your ability to identify edges, endpoints, and accumulation behavior. By the end, you’ll understand exactly how to locate boundary points and explain what they reveal about a set in topology!
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2) What first name or nickname would you like us to use?

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2) Which are boundary points of the interval (0,1)?

0.5

0 and 1 are boundary points because every interval around them hits both inside (0,1) and outside it.

Explanation

0 and 1 are boundary points because every interval around them hits both inside (0,1) and outside it.

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3) Which statements are true about boundary points?

They are always limit points

They are always interior points

They can lie inside or outside the set

They are points on the edge of the set

Boundary points:

are always limit points (A),

can lie inside or outside the set (C),

lie on the “edge” of the set (D).

They are not interior points.

Explanation

Boundary points:

are always limit points (A),

can lie inside or outside the set (C),

lie on the “edge” of the set (D).

They are not interior points.

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4) Which sets have an empty boundary?

ℝ

∅

(0,1)

{1}

ℝ has no outside points, and ∅ has no points at all. Both have empty boundary.

Explanation

ℝ has no outside points, and ∅ has no points at all. Both have empty boundary.

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5) Which formulas correctly represent boundary(A)?

Cl(A) − int(A)

Cl(A) ∩ int(A)

A − Aᶜ

A ∪ Aᶜ

Boundary(A) = closure(A) − interior(A).

Explanation

Boundary(A) = closure(A) − interior(A).

6) Points 0 and 1 are boundary points of:

(0,1)

[0,1]

{0,1}

(-1,2)

Both 0 and 1 lie inside (−1,2) but each neighborhood touches outside too, so they are boundary points.

Explanation

Both 0 and 1 lie inside (−1,2) but each neighborhood touches outside too, so they are boundary points.

7) Boundary(A) is always:

Closed

Open

A subset of cl(A)

Equal to A

The boundary is always a closed set and is always contained within the closure of A.

Explanation

The boundary is always a closed set and is always contained within the closure of A.

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8) Which sets have boundary {0}?

[0,∞)

(0,∞)

(-∞,0]

(-∞,0)

For [0,∞), the only edge point is 0.

For (−∞,0], again only 0 is the boundary.

Explanation

For [0,∞), the only edge point is 0.

For (−∞,0], again only 0 is the boundary.

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9) Which sets have infinite boundaries?

ℚ

{1/n : n∈ℕ}

(0,1)

The Cantor set

ℚ is dense everywhere, so its boundary is all real numbers—an infinite set.

The Cantor set’s boundary is the set itself, which is infinite.

Explanation

ℚ is dense everywhere, so its boundary is all real numbers—an infinite set.

The Cantor set’s boundary is the set itself, which is infinite.

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10) Identify boundary of (0,1) ∪ (2,3):

Each endpoint is a point where neighborhoods touch inside the set and outside the set.

Explanation

Each endpoint is a point where neighborhoods touch inside the set and outside the set.

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11) A point cannot be both:

Boundary and limit point

Boundary and interior

Closure and boundary

Limit and exterior

Interior points require neighborhoods fully inside A; boundary points require touching outside. Both cannot occur together.

Explanation

Interior points require neighborhoods fully inside A; boundary points require touching outside. Both cannot occur together.

12) A set whose closure equals its boundary must:

Have an empty interior

Be closed

Be perfect

Be nowhere dense

If closure = boundary, then interior must be empty.

Explanation

If closure = boundary, then interior must be empty.

13) A set A whose closure has empty interior must have:

No boundary

Boundary equal to cl(A)

Boundary possibly equal to A

Boundary infinite

If interior is empty, then boundary(A) = closure(A).

Explanation

If interior is empty, then boundary(A) = closure(A).

14) Identify boundary of A = {1/n : n∈ℕ} ∪ [2,3].

{1/n : n∈ℕ}

1/n approaches 0 → boundary at 0.

[2,3] has boundary points 2 and 3.

Explanation

1/n approaches 0 → boundary at 0.

[2,3] has boundary points 2 and 3.

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15) If A is perfect (closed and has no isolated points), boundary(A) is:

∅

ℝ

Cl(Aᶜ)

A perfect set is equal to its boundary because every point is a limit point and the set is closed.

Explanation

A perfect set is equal to its boundary because every point is a limit point and the set is closed.

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