Advanced Compactness and Open Cover Applications

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Quizzes Created: 7060 | Total Attempts: 9,520,935
| Questions: 14 | Updated: Oct 13, 2025
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1) Every closed interval in ℝ is compact.

Explanation

Heine–Borel theorem.

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About This Quiz
Advanced Compactness And Open Cover Applications - Quiz

Beyond the basics! This quiz brings complex problems and abstract reasoning about open covers. Try this quiz to challenge your grasp of compactness.

2)
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2) Which is compact?

Explanation

Closed and bounded.

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3) Which open cover demonstrates compactness of [0,1]?

Explanation

Finite subcover exists.

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4) A bounded open interval is compact.

Explanation

Open intervals are not compact.

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5) Which is not compact in ℝ?

Explanation

Open intervals not compact.

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6) Which property ensures compactness in ℝ^n?

Explanation

Closed and bounded sets are compact.

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7) Which is an open cover of {0}?

Explanation

Any open set containing 0 works.

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8) Compactness guarantees:

Explanation

Definition of compactness.

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9) ℕ is compact in ℝ.

Explanation

ℕ is not bounded.

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10) Which is compact in ℝ?

Explanation

Finite union of compact sets is compact.

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11) Which is a counterexample to compactness?

Explanation

No finite subcover.

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12) The union of two compact sets is compact.

Explanation

Union of compact sets is compact.

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13) Which open cover proves {1/n : n∈ℕ} not compact without 0?

Explanation

Needs infinitely many sets.

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14) Which statement is true?

Explanation

Heine–Borel theorem says compactness in ℝ means closed and bounded.

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Every closed interval in ℝ is compact.
Which is compact?
Which open cover demonstrates compactness of [0,1]?
A bounded open interval is compact.
Which is not compact in ℝ?
Which property ensures compactness in ℝ^n?
Which is an open cover of {0}?
Compactness guarantees:
ℕ is compact in ℝ.
Which is compact in ℝ?
Which is a counterexample to compactness?
The union of two compact sets is compact.
Which open cover proves {1/n : n∈ℕ} not compact without 0?
Which statement is true?
Alert!

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