Interior Points Quiz

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Quizzes Created: 7288 | Total Attempts: 9,526,295
| Questions: 15 | Updated: Nov 24, 2025
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1) An interior of a set U is a point for which there exists an open ball completely contained in U.

Explanation

This is the definition of an interior point—there must be a small open ball inside the set.

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About This Quiz
Interior Points Quiz - Quiz

Ready to dig deeper into the structure of sets in topology? This quiz helps you understand what interior points are and how they determine the “inside” of a set. You’ll work with open balls, intervals, and common set operations to see when a point qualifies as interior. From analyzing unions... see moreand intersections to comparing interior and boundary behavior, you’ll build a clear understanding of how mathematicians describe the inside of a space. By the end, you’ll be able to recognize interior points confidently and understand why they matter in topology! see less

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2) Every interior of a set U must also be a boundary point of that set.

Explanation

Interior points are inside the set, boundary points are on the 'edge.' An interior point is not a boundary point.

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3) The interior of a set U is always an open set.

Explanation

The interior is defined as the largest open set inside U, so it is always open.

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4) If set U is open, then every point in the set is an interior point.

Explanation

In an open set, every point has a small open interval around it inside the set.

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5) If set U is closed, then it cannot contain any interior points.

Explanation

A set can be closed and still have interior points—for example, the interval [0,1] has interior (0,1).

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6) The empty set ∅ has no interior points.

Explanation

There are no points in ∅, so it has no interior points.

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7) A point can be an interior point of U even if it lies outside of U.

Explanation

Interior points must lie inside the set.

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8) Which statement is true?

Explanation

The interior of a union is at least as big as the union of interiors.

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9) If A is closed and has a nonempty interior, then:

Explanation

Having a nonempty interior means A includes some open interval.

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10) Which set has a nonempty interior?

Explanation

(0,1) is an open interval, so it has interior points.

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11) If x is an interior point of A and A ⊆ B, then:

Explanation

If a ball around x is inside A, it is also inside B, so x is interior to B.

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12) Which statement is true?

Explanation

The interior of an intersection is the intersection of the interiors.

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13) Suppose A is open and int(B) ⊆ A, then:

Explanation

If int(B) ⊆ A, then B has at least some open part that lies inside A.

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14) The statement int(A) ⊆ A is:

Explanation

The interior of a set is always made of points inside that set.

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15) A point is not an interior point if:

Explanation

If no open ball stays inside the set, then the point is not interior.

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An interior of a set U is a point for which there exists an open ball...
Every interior of a set U must also be a boundary point of that set.
The interior of a set U is always an open set.
If set U is open, then every point in the set is an interior point.
If set U is closed, then it cannot contain any interior points.
The empty set ∅ has no interior points.
A point can be an interior point of U even if it lies outside of U.
Which statement is true?
If A is closed and has a nonempty interior, then:
Which set has a nonempty interior?
If x is an interior point of A and A ⊆ B, then:
Which statement is true?
Suppose A is open and int(B) ⊆ A, then:
The statement int(A) ⊆ A is:
A point is not an interior point if:
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