Boundary Points 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) A boundary point of a set is a point where every open interval around it contains points from the set.

Explanation

Boundary points require every neighborhood to meet the set.

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About This Quiz
Boundary Points Quiz - Quiz

Ready to understand the “edge” of a set in topology? This quiz guides you through the concept of boundary points, where every neighborhood touches both a set and its complement. You’ll examine boundaries of intervals, unions, open and closed sets, and classic sets like ℚ. Through these questions, you’ll build... see morean understanding of how boundaries reveal the structure of a set and how they relate to closure and interior. By the end, you’ll be able to identify boundary points confidently and understand why they play an essential role in topology!
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2) Boundary points must always lie inside the set.

Explanation

They can be inside or outside.

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3) Every open interval has no boundary points.

Explanation

An open interval has boundary points—its endpoints.

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4) If A is closed, then all of its points are boundary points.

Explanation

Closed intervals have interior points too.

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5) The boundary of a closed set is always closed.

Explanation

Boundaries are always closed.

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6) The boundary of a closed interval [a,b] is the set {a,b}.

Explanation

The endpoints are exactly its boundary.

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7) The empty set has an empty boundary.

Explanation

No points = no boundary points.

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8) If A is open, then:

Explanation

Open sets can have boundaries outside them.

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9) If A is closed, which is true?

Explanation

Closed sets contain all their boundary points.

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10) The boundary of ℚ is:

Explanation

Rationals are dense; boundary is all real numbers.

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11) The boundary of a union of disjoint intervals can be:

Explanation

Depends on how many disjoint intervals there are.

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12) If cl(A) = int(A), then boundary(A) is:

Explanation

Boundary = closure – interior = ∅.

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13) A point x is a boundary point of A iff:

Explanation

Boundary = cl(A) ∩ cl(Aᶜ).

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14) If A has an empty boundary, which must be true?

Explanation

Only ∅ and ℝ have no boundary.

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15) The boundary of A ∪ B equals boundary(A) ∪ boundary(B) only if:

Explanation

Works only when boundaries do not create new boundary points.

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A boundary point of a set is a point where every open interval around...
Boundary points must always lie inside the set.
Every open interval has no boundary points.
If A is closed, then all of its points are boundary points.
The boundary of a closed set is always closed.
The boundary of a closed interval [a,b] is the set {a,b}.
The empty set has an empty boundary.
If A is open, then:
If A is closed, which is true?
The boundary of ℚ is:
The boundary of a union of disjoint intervals can be:
If cl(A) = int(A), then boundary(A) is:
A point x is a boundary point of A iff:
If A has an empty boundary, which must be true?
The boundary of A ∪ B equals boundary(A) ∪ boundary(B) only if:
Alert!