Open Sets Properties Quiz

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Quizzes Created: 7387 | Total Attempts: 9,527,684
| Questions: 15 | Updated: Nov 24, 2025
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1) Which of the following is always open in any topology?

Explanation

The empty set is always open in every topology by definition.

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About This Quiz
Open Sets Properties Quiz - Quiz

Think you know how open sets behave? This undergraduate-level quiz takes you deeper into the properties that define open sets in topology. You’ll examine how operations like unions and intersections affect openness, test examples from ℝ, and explore subtle cases involving sequences and boundary points. These questions help you sharpen... see moreyour ability to recognize when a set is truly open and understand the logic behind topological structures. By the end, you’ll feel more confident applying topology rules to real mathematical problems. see less

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2) The intersection of two open sets is:

Explanation

The intersection of two open sets is always open.

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3) Which of the following is open in ℝ?

Explanation

(−∞,5) is an open interval. The other sets include boundary points.

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4) Which of the following is open in ℝ?

Explanation

(0,1) excludes both endpoints, so it is open.

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5) A set that is both open and closed is called:

Explanation

A “clopen” set is both open and closed.

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6) The union of (0,1) and (2,3) is:

Explanation

(0,1) and (2,3) are open intervals. Their union is also open.

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7) The intersection of (−1,1) and (0,2) is:

Explanation

The intersection is (0,1), which is open.

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8) The union of (0,1] and (2,3) is:

Explanation

(0,1] is not open, so the union is not open; it's also not closed.

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9) In ℝ, the set {1/n : n ∈ ℕ} is:

Explanation

The set has limit point 0 (not included), so it is not closed, and it's not open. Hence neither.

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10) Which is an open set in ℝ?

Explanation

(1,2) and (3,4) are open intervals; their union is open.

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11) Set B = (1,−2] ∪ [3,4) is an open set.

Explanation

Both intervals include endpoints, so the set is not open.

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12) A set with no interior points cannot be open.

Explanation

Open sets must contain at least one interior point.

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13) Set C = {x ∈ ℝ : x² + 1 < 2} is an open set.

Explanation

x² + 1

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14) If every sequence in set A that converges in the space has its limits inside A, then A must be open.

Explanation

This condition describes a closed set, not an open one.

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15) The set A = {x ∈ ℝ : x² < 9} is an open set.

Explanation

x²

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Which of the following is always open in any topology?
The intersection of two open sets is:
Which of the following is open in ℝ?
Which of the following is open in ℝ?
A set that is both open and closed is called:
The union of (0,1) and (2,3) is:
The intersection of (−1,1) and (0,2) is:
The union of (0,1] and (2,3) is:
In ℝ, the set {1/n : n ∈ ℕ} is:
Which is an open set in ℝ?
Set B = (1,−2] ∪ [3,4) is an open set.
A set with no interior points cannot be open.
Set C = {x ∈ ℝ : x² + 1 < 2} is an open set.
If every sequence in set A that converges in the space has its limits...
The set A = {x ∈ ℝ : x² < 9} is an open set.
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