Heine–Borel Compactness Concepts Quiz

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Quizzes Created: 7387 | Total Attempts: 9,541,629
| Questions: 15 | Updated: Dec 12, 2025
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1) If a set in ℝ is bounded but not closed, it cannot be compact.

Explanation

In ℝ, compact ⇔ closed and bounded. Missing closedness ⇒ not compact.

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About This Quiz
Heineborel Compactness Concepts Quiz - Quiz

How well can you spot compactness using the Heine–Borel theorem? This quiz guides you through determining whether sets are compact by checking whether they are closed, bounded, or missing limit points. You’ll evaluate intervals, discrete sets, and infinite unions, and identify which sets fail compactness using open covers. Through these... see moreexamples, you’ll deepen your understanding of sequential compactness and finite-subcover behavior. By the end, you’ll confidently apply Heine–Borel to classify sets and explain why compactness matters in real analysis!
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2) The open interval (0,1) is not compact because it fails to contain all of its limit points.

Explanation

(0,1) is missing 0 and 1, so it is not closed → not compact.

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3) A finite union of compact sets in ℝ is compact.

Explanation

Compactness is preserved under finite unions.

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4) [0,1] ∪ (2,3) is compact because it is bounded, even though it is not closed.

Explanation

[0,1] ∪ (2,3) is not closed, so it is not compact.

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5) Every open cover of a compact set in ℝ must contain a finite subcover.

Explanation

This is the definition of compactness.

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6) Which of the following sets is compact in ℝ?

Explanation

Only [0,1] is both closed and bounded.

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7) Which of the following is not compact, even though it is closed?

Explanation

ℤ is closed but unbounded.

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8) Which set has open cover with no finite subcover?

Explanation

(0,1) not compact.

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9) A continuous function on compact subset must attain max/min.

Explanation

Extreme Value Theorem.

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10) Condition for compactness in ℝ²?

Explanation

Heine-Borel.

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11) Which sets are compact in ℝ?

Explanation

[0,1] and {1/n}∪{0} are closed and bounded.

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12) Which conditions guarantee not compact?

Explanation

All break compactness.

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13) Closed and bounded hence compact?

Explanation

(0,1] not closed.

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14) Sequential compact sets?

Explanation

Sequential compact ⇔ compact.

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15) Noncompact sets via open cover?

Explanation

All noncompact.

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If a set in ℝ is bounded but not closed, it cannot be compact.
The open interval (0,1) is not compact because it fails to contain all...
A finite union of compact sets in ℝ is compact.
[0,1] ∪ (2,3) is compact because it is bounded, even though it is...
Every open cover of a compact set in ℝ must contain a finite...
Which of the following sets is compact in ℝ?
Which of the following is not compact, even though it is closed?
Which set has open cover with no finite subcover?
A continuous function on compact subset must attain max/min.
Condition for compactness in ℝ²?
Which sets are compact in ℝ?
Which conditions guarantee not compact?
Closed and bounded hence compact?
Sequential compact sets?
Noncompact sets via open cover?
Alert!

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