Heine–Borel Theorem Quiz

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Thames
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| Questions: 15 | Updated: Dec 12, 2025
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1) In ℝⁿ with the usual topology, a set is compact if and only if it is closed and bounded.

Explanation

The Heine–Borel theorem states exactly this for ℝⁿ.

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About This Quiz
Heineborel Theorem Quiz - Quiz

Ready to understand one of the most powerful results in real analysis? This quiz focuses on the Heine–Borel theorem, which characterizes compact sets in ℝⁿ as exactly those that are closed and bounded. You’ll test examples such as intervals, discrete sets, and unbounded subsets to see which are compact and... see morewhy. Through these questions, you’ll explore how compactness guarantees extrema, finite subcovers, and completeness. By the end, you’ll confidently apply the Heine–Borel theorem to determine compactness in real spaces!
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2) The Heine–Borel Theorem applies to all topological spaces.

Explanation

Heine–Borel only holds in Euclidean spaces.

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3) The interval [0,1] is compact in ℝ because it is closed and bounded.

Explanation

Closed + bounded ⇒ compact.

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4) The set (0,1) is not compact in ℝ because it is not closed.

Explanation

It is bounded but not closed.

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5) A closed interval [a,b] is compact even if a > b.

Explanation

If a>b interval empty; empty sets are compact but expression invalid.

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6) Boundedness alone is sufficient for compactness in ℝ.

Explanation

Must also be closed.

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7) Closedness alone is sufficient for compactness in ℝ.

Explanation

ℤ is closed but unbounded.

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8) Every compact subset of ℝ must have a finite maximum and minimum value.

Explanation

Continuous identity attains max/min.

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9) Any finite union of compact subsets of ℝ is compact.

Explanation

Finite union compact.

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10) Heine–Borel guarantees every open cover of closed+bounded set has finite subcover.

Explanation

Exactly Heine–Borel.

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

Explanation

Only [0,1] is closed+bounded.

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12) Which set is not bounded and therefore not compact in ℝ?

Explanation

[1,∞) unbounded.

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13) By Heine–Borel, which condition guarantees compactness in ℝⁿ?

Explanation

Closed+bounded.

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14) Which set is not compact even though closed?

Explanation

ℤ closed but unbounded.

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15) Which set must contain sup & inf because compact?

Explanation

Cantor set compact.

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In ℝⁿ with the usual topology, a set is compact if and only if it...
The Heine–Borel Theorem applies to all topological spaces.
The interval [0,1] is compact in ℝ because it is closed and bounded.
The set (0,1) is not compact in ℝ because it is not closed.
A closed interval [a,b] is compact even if a > b.
Boundedness alone is sufficient for compactness in ℝ.
Closedness alone is sufficient for compactness in ℝ.
Every compact subset of ℝ must have a finite maximum and minimum...
Any finite union of compact subsets of ℝ is compact.
Heine–Borel guarantees every open cover of closed+bounded set has...
Which of the following sets is compact in ℝ?
Which set is not bounded and therefore not compact in ℝ?
By Heine–Borel, which condition guarantees compactness in ℝⁿ?
Which set is not compact even though closed?
Which set must contain sup & inf because compact?
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