What Makes a Set Compact?

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| Questions: 15 | Updated: Oct 13, 2025
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1) Which sequence property is equivalent to compactness in ℝ?

Explanation

Sequential compactness is equivalent to compactness in ℝ.

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About This Quiz
What Makes A Set Compact? - Quiz

Not all sets are compact. In this quiz, you’ll explore the definition and learn what truly makes a set compact. Take this quiz to get clarity on compactness.

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2) Every continuous function on a compact set attains a maximum and minimum.

Explanation

By the extreme value theorem, continuous functions reach maxima/minima on compact sets.

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3) Which of these functions attains both max and min on [0,1]?

Explanation

sin(x) is continuous on [0,1] and achieves both maximum and minimum.

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4) Which of these sets is compact in ℝ²?

Explanation

Closed and bounded subsets of ℝ² are compact.

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5) In ℝⁿ, a closed and bounded set is compact.

Explanation

Heine–Borel theorem generalizes to ℝⁿ.

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6) Which statement about compactness is true?

Explanation

Compactness and sequential compactness coincide in metric spaces like ℝ.

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7) Which property is preserved under continuous maps?

Explanation

The continuous image of a compact set is compact.

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8) Example: Which is compact under f(x)=x² mapping?

Explanation

[0,1] is compact, and its image under x² is also compact.

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9) The product of two compact sets is compact.

Explanation

Finite product of compact sets is compact (Tychonoff’s theorem).

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10) Which is compact in ℝ?

Explanation

Adding 0 makes the set closed and bounded, hence compact.

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11) Which theorem guarantees continuous functions on compact sets are uniformly continuous?

Explanation

Heine–Cantor ensures uniform continuity on compact sets.

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12) Compact subsets of metric spaces are always closed.

Explanation

In metric spaces, compact sets are closed and bounded.

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13) Which is not compact in ℝ³?

Explanation

ℝ³ is unbounded, so it cannot be compact.

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14) Which property is NOT guaranteed by compactness?

Explanation

Compactness requires closed + bounded, not openness.

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15) In general topological spaces, compact ≠ closed and bounded.

Explanation

Closed + bounded implies compact only in ℝⁿ, not in general spaces.

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Which sequence property is equivalent to compactness in ℝ?
Every continuous function on a compact set attains a maximum and...
Which of these functions attains both max and min on [0,1]?
Which of these sets is compact in ℝ²?
In ℝⁿ, a closed and bounded set is compact.
Which statement about compactness is true?
Which property is preserved under continuous maps?
Example: Which is compact under f(x)=x² mapping?
The product of two compact sets is compact.
Which is compact in ℝ?
Which theorem guarantees continuous functions on compact sets are...
Compact subsets of metric spaces are always closed.
Which is not compact in ℝ³?
Which property is NOT guaranteed by compactness?
In general topological spaces, compact ≠ closed and bounded.
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