Heine–Borel Applications Quiz

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| By Thames
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Thames
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Quizzes Created: 7387 | Total Attempts: 9,537,848
| Questions: 15 | Updated: Dec 12, 2025
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1) The set ([0,1) ∪ (1,2]) is compact in ℝ.

Explanation

The set is not closed because it excludes 1, so it cannot be compact in ℝ.

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About This Quiz
Heineborel Applications Quiz - Quiz

Think you can apply the Heine–Borel theorem confidently? This quiz helps you analyze sequences, subspaces, and various closed and bounded sets to determine compactness. You’ll work through examples such as intervals, unions, and sets in ℝ², seeing how compactness ensures convergent subsequences and function extrema. These problems will strengthen you... see moreunderstanding of how closedness and boundedness interact with topology. By the end, you’ll be ready to identify compact sets and apply the Heine–Borel theorem across multiple scenarios!
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2) A sequence in a compact subset of ℝ always contains a convergent subsequence.

Explanation

Bolzano–Weierstrass theorem.

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3) The Heine–Borel theorem implies that ℝ is not compact.

Explanation

ℝ is unbounded.

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4) The set of natural numbers ℕ is compact.

Explanation

ℕ is closed but unbounded.

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5) If a subset of ℝ is compact, then it is contained in some closed interval.

Explanation

Compact sets are bounded.

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6) Open cover showing (0,1] not compact

Explanation

Does not cover near 0 finitely.

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7) Which subset is compact?

Explanation

Finite union of compact sets.

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8) Continuous f on compact K attains max/min

Explanation

Extreme Value Theorem.

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9) Which set is compact in ℝ²?

Explanation

Closed and bounded.

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10) Which property fails for (0,1]?

Explanation

Not closed.

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11) Heine-Borel application

Explanation

Heine–Borel.

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12) Closed+bounded but not compact

Explanation

Infinite dimensional.

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13) Compact subset of ℝ must be

Explanation

Heine–Borel.

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14) Set with open cover w/o finite subcover

Explanation

Not compact.

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15) Compact sets in ℝ

Explanation

Compact ⇒ closed & bounded.

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The set ([0,1) ∪ (1,2]) is compact in ℝ.
A sequence in a compact subset of ℝ always contains a convergent...
The Heine–Borel theorem implies that ℝ is not compact.
The set of natural numbers ℕ is compact.
If a subset of ℝ is compact, then it is contained in some closed...
Open cover showing (0,1] not compact
Which subset is compact?
Continuous f on compact K attains max/min
Which set is compact in ℝ²?
Which property fails for (0,1]?
Heine-Borel application
Closed+bounded but not compact
Compact subset of ℝ must be
Set with open cover w/o finite subcover
Compact sets in ℝ
Alert!

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