Basics of Open Sets Quiz

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| Questions: 15 | Updated: Nov 24, 2025
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1) Every open interval in the real numbers, such as (2,5), is an open set in the standard topology on ℝ.

Explanation

True, an open interval like (2,5) does not include its endpoints, so every point has a small interval around it that stays inside, which makes it open.

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About This Quiz
Open Sets Quizzes & Trivia

Are you ready to build confidence in foundational topology? Take this undergraduate-level quiz where you’ll explore what makes a set “open” in mathematics. You’ll work with ideas like interior points, open intervals, unions, intersections, and complements — just like the concepts used in real analysis and higher-level math. By practicing... see morewith examples in ℝ and ℝ², you’ll learn how to decide whether a set is open and why these definitions matter. Get ready to strengthen your understanding of how topology shapes the structure of mathematical spaces! see less

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2) The union of any collection of open sets is always an open set.

Explanation

If a point is in the union, it belongs to at least one open set, so it has a neighborhood inside that set and therefore the union is open.

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3) The intersection of two open sets is always an open set.

Explanation

A point in both open sets has a small neighborhood inside each one; a smaller neighborhood stays in the intersection, so the intersection is open.

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4) The empty set ∅ is considered an open set.

Explanation

By definition in topology, the empty set is always an open set.

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5) An interval [1,3] is considered an open set.

Explanation

[1,3] includes its endpoints, so you cannot place a small interval around 1 or 3 that stays inside the set; therefore it is not open.

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6) A set is open if it does not contain its boundary points.

Explanation

In ℝ, open sets exclude all of their boundary points; that is a basic characterization of openness.

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7) The complement of a closed set is an open set.

Explanation

In topology, a set is closed if and only if its complement is open, so the complement of a closed set is open.

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8) A set that is not closed must be open.

Explanation

A set can be neither open nor closed; for example, [0,1) in ℝ is not open and not closed.

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9) If all the boundary points are included in the set, then it is an open set.

Explanation

If all boundary points are included, the set is closed, not open.

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10) Set A = {x : −3 < x < 3, x ∈ ℝ} is an open set.

Explanation

This is the open interval (−3,3), which does not include its endpoints, so it is an open set.

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11) Which of the following can be both open and closed?

Explanation

The empty set is one of the special sets (along with the whole space) that is both open and closed (clopen).

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12) Which of the following statements about open sets is false?

Explanation

Countable intersections of open sets need not be open; for example, the intersection of (−1/n, 1/n) over n gives {0}, which is not open in ℝ.

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13) A set U in a topological space (X,τ) is called open if:

Explanation

A set is open if every point in it is an interior point, meaning each point has a neighborhood contained entirely in U.

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14) In ℝ², the set {(x,y): x² + y² < 1} is:

Explanation

This is the interior of the unit disk; it does not include the boundary circle x² + y² = 1, so it is an open set.

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15) If U is open in ℝ under the standard topology, then:

Explanation

Every open set in ℝ can be written as a union (possibly infinite) of open intervals.

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Every open interval in the real numbers, such as (2,5), is an open set...
The union of any collection of open sets is always an open set.
The intersection of two open sets is always an open set.
The empty set ∅ is considered an open set.
An interval [1,3] is considered an open set.
A set is open if it does not contain its boundary points.
The complement of a closed set is an open set.
A set that is not closed must be open.
If all the boundary points are included in the set, then it is an open...
Set A = {x : −3 < x < 3, x ∈ ℝ} is an open set.
Which of the following can be both open and closed?
Which of the following statements about open sets is false?
A set U in a topological space (X,τ) is called open if:
In ℝ², the set {(x,y): x² + y² < 1} is:
If U is open in ℝ under the standard topology, then:
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