Boundary Points Properties Quiz

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Thames
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Quizzes Created: 8157 | Total Attempts: 9,566,492
| Attempts: 13 | Questions: 15 | Updated: Dec 12, 2025
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Question 1 / 16
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1) Boundary points can be endpoints of the interval.

Explanation

Endpoints like 0 and 1 in (0,1) or [0,1] meet the definition of boundary points.

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About This Quiz
Boundary Points Properties Quiz - Quiz

Think you understand boundary points? This quiz explores how boundaries behave in different types of sets, from open intervals to half-lines and infinite sets. You’ll test your understanding of when a point is a boundary point, how boundaries interact with closure, and whether boundaries can lie inside or outside a... see moreset. These questions help build your intuition about the “edges” of sets in ℝ and other topological spaces. By the end, you’ll feel confident explaining and identifying boundary points in real-world and mathematical situations!
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2)
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2) The boundary of the interval [0,∞) is {0}.

Explanation

Only 0 touches both the interval and the outside (negative numbers). Points >0 do not touch anything outside.

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3) If A has no limit points, then the boundary of A is empty.

Explanation

Boundary points are always limit points or isolated points that still touch the complement; if no limit points exist, there is no boundary.

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4) If a point is in the closure of Aᶜ, it must be a boundary point of A.

Explanation

A boundary point must be in both closure(A) and closure(Aᶜ).

Being only in the closure of Aᶜ is not enough.

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5) The boundary of an open interval (0,1):

Explanation

The points 0 and 1 are not in the interval, but every neighborhood of 0 or 1 touches the interval and the outside, so the boundary is {0,1}.

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6) The boundary of a closed interval [0,1]:

Explanation

The endpoints 0 and 1 are where neighborhoods meet both the inside and outside of the interval, so the boundary is {0,1}.

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7) The boundary of (1,∞):

Explanation

Only the point 1 is a boundary point because intervals around 1 touch both (1,∞) and numbers less than 1.

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8) Which statement best describes a boundary point of a set A?

Explanation

A boundary point is one where every neighborhood touches both the set and its complement, meaning no matter how small the interval, it contains points inside A and outside A.

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9) Boundary points of a set can be:

Explanation

A boundary point may lie inside the set or outside the set, as long as every neighborhood intersects both A and its complement.

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10) The boundary of ℝ:

Explanation

ℝ has no “outside,” so no point can touch both ℝ and its complement. The boundary is ∅.

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11) A boundary point is always:

Explanation

Boundary points are always limit points, since they require neighborhoods that always meet the set.

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12) A set with no interior points must have an empty boundary.

Explanation

Example: ℚ has empty interior but boundary ℝ.

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13) A point can be both an interior point and a boundary point simultaneously.

Explanation

Interior points require neighborhoods fully inside the set; boundary points require neighborhoods touching outside, so both cannot occur together.

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14) The boundary of the union of two sets is always the union of their boundaries.

Explanation

This is not always true—taking the union can create new boundary points or remove some.

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15) The boundary of the intersection of two sets is always the intersection of their boundaries.

Explanation

This fails in general because intersections may change which points lie on the edge.

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Boundary points can be endpoints of the interval.
The boundary of the interval [0,∞) is {0}.
If A has no limit points, then the boundary of A is empty.
If a point is in the closure of Aᶜ, it must be a boundary point of...
The boundary of an open interval (0,1):
The boundary of a closed interval [0,1]:
The boundary of (1,∞):
Which statement best describes a boundary point of a set A?
Boundary points of a set can be:
The boundary of ℝ:
A boundary point is always:
A set with no interior points must have an empty boundary.
A point can be both an interior point and a boundary point...
The boundary of the union of two sets is always the union of their...
The boundary of the intersection of two sets is always the...
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