Identifying Open Sets Quiz

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1) Which of the following correctly describes an open set?

A set U is open if it contains all its boundary points

A set U is open if its complement is closed

A set U is open if it contains at least one isolated point

Open means every point has a small neighborhood fully inside the set.

Explanation

Open means every point has a small neighborhood fully inside the set.

About This Quiz

Ready to apply your understanding of open sets in real situations? This undergraduate-level quiz helps you practice recognizing open sets using formal definitions and intuitive reasoning. You’ll identify open sets using interior points, neighborhoods, complements, and interval behavior — and explore how these ideas show up in everyday contexts like... see moretime intervals, temperatures, speeds, and depths. By the end, you'll be able to decide quickly and confidently whether a set is open, both mathematically and conceptually. see less

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2) Which of the following statements correctly identify an open set?

An open set contains no end points of any interval it includes

An open set is any set where each point has a neighborhood fully contained within the set

An open set is any set that includes at least one open interval

A set is open if it can be expressed as a union of open intervals

Open sets must give each point its own neighborhood. They can also be expressed as unions of open intervals.

Explanation

Open sets must give each point its own neighborhood. They can also be expressed as unions of open intervals.

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3) Which definitions correctly describe open sets in a topological space?

A set U is open if every point in U is an interior point

A set U is open if it contains no interior point

A set U is open if its complement is closed in the same topology

A set U is open if it is a finite intersection of closed sets

Open means all points are interior points. Also, the complement of a closed set is open.

Explanation

Open means all points are interior points. Also, the complement of a closed set is open.

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4) Which of the following are open in ℝ?

∅

∩ₙ₌₁¹⁰ (−1/n, 1/n)

∩ₙ₌₁∞ (−1/n, 1/n)

∪ₙ=1∞ (1/n, 1)

∅ is always open. Finite intersections of open intervals are open. A union of open intervals is open. The infinite intersection in C gives {0}, which is not open.

Explanation

∅ is always open. Finite intersections of open intervals are open. A union of open intervals is open. The infinite intersection in C gives {0}, which is not open.

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5) Which sets are open?

(0,1) ∪ [2,3]

(0,1) ∪ (2,3)

ℝ{0}

[0,1)

A union of open intervals is open. ℝ without a single point is also open.

Explanation

A union of open intervals is open. ℝ without a single point is also open.

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6) Which of the following are open in ℝ?

(0,2) ∪ (1,3)

∩ₙ₌₁∞ (0, 1 + 1/n)

∩ₙ₌₁∞ (0, 1/n)

(−∞,0) ∪ (1,∞)

Unions and finite intersections of open intervals remain open. The intersection in C collapses to {0}, not open.

Explanation

Unions and finite intersections of open intervals remain open. The intersection in C collapses to {0}, not open.

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7) Identify open sets.

[0,1] ∪ (1,2)

(0,1] ∪ (1,2)

(0,1) ∪ (2,3) ∪ (4,5)

∅

A union of open intervals is open. ∅ is always open.

Explanation

A union of open intervals is open. ∅ is always open.

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8) Which of the following are open?

{x | x² < 4}

{x | x² ≤ 4}

ℝ \ [0,1]

ℝ \ (0,1]

x²

Explanation

x²

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9) Consider these sets. Which are open?

∪ₙ=1∞ (n, n+1)

∩ₙ=1∞ (0, 1 + 1/n)

∩ₙ=1∞ (0, 1/n)

(−∞,0]

A union of open intervals is open. The intersection in B is (0,1), which is open.

Explanation

A union of open intervals is open. The intersection in B is (0,1), which is open.

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10) Which of the following are open sets?

{x | 1 < x < 2}

{x | 1 ≤ x < 2}

{x | x ≠ 0}

{0}

(1,2) is open. ℝ without a single point is also open.

Explanation

(1,2) is open. ℝ without a single point is also open.

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11) Identify all open sets.

ℝ \ ℤ

ℝ \ {1,2,3}

(−∞,0) ∪ (3,∞)

(−∞,0) ∪ [3,∞)

Removing isolated points or integers from ℝ leaves an open set. Unions of open intervals are open.

Explanation

Removing isolated points or integers from ℝ leaves an open set. Unions of open intervals are open.

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12) Which of the following sets are open in ℝ?

The set of all temperatures strictly between 20℃ to 30℃.

The set of all temperatures including 20℃ but strictly less than 30℃.

The set of all temperatures less than or equal to 25℃.

The set of all real numbers except freezing point, 0℃.

Strict inequalities give open intervals. Removing a single point also gives an open set.

Explanation

Strict inequalities give open intervals. Removing a single point also gives an open set.

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13) Which of the following sets of time intervals are open?

All time strictly between 8:00 AM to 10:00 AM

Times between 8:00 AM inclusive to 10:00 AM exclusive

All times outside lunch hour

All times between 6:00 AM and 8:00 AM, including 6:00 AM

Strict inequality gives an open set. "Outside a closed interval" is also open.

Explanation

Strict inequality gives an open set. "Outside a closed interval" is also open.

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14) Which sets representing water depth are open?

Depths strictly between 2m and 5m

Depths from 2m inclusive and 5m exclusive

Depth of 1m only

Depths not including exactly 3m

Strict intervals are open. Removing one point (depth ≠ 3) gives an open set.

Explanation

Strict intervals are open. Removing one point (depth ≠ 3) gives an open set.

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15) Identify open sets of allowed speeds on a highway in kph

Speeds including 60 but below 120

Speeds strictly between 60 and 120

Speeds 120 or lower

Speeds excluding exactly 80

Strict intervals are open. Removing a single point also results in an open set.

Explanation

Strict intervals are open. Removing a single point also results in an open set.

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Identifying Open Sets Quiz

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