What Makes a Set Compact?

1) Which sequence property is equivalent to compactness in ℝ?

Every sequence diverges

Every sequence has a convergent subsequence

Every sequence is bounded

Every sequence is finite

Sequential compactness is equivalent to compactness in ℝ.

Explanation

Sequential compactness is equivalent to compactness in ℝ.

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

True

False

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

Explanation

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

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

F(x)=1/x

F(x)=sin(x)

F(x)=1/(x−1)

F(x)=tan(x)

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

Explanation

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

4) Which of these sets is compact in ℝ²?

The closed unit disk {x²+y² ≤ 1}

The open disk {x²+y² < 1}

ℝ²

The half-plane {x ≥ 0}

Closed and bounded subsets of ℝ² are compact.

Explanation

Closed and bounded subsets of ℝ² are compact.

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

True

False

Heine–Borel theorem generalizes to ℝⁿ.

Explanation

Heine–Borel theorem generalizes to ℝⁿ.

6) Which statement about compactness is true?

Every compact set is finite

Every compact set in ℝ is sequentially compact

Compactness only applies to intervals

Compact sets must be countable

Compactness and sequential compactness coincide in metric spaces like ℝ.

Explanation

Compactness and sequential compactness coincide in metric spaces like ℝ.

7) Which property is preserved under continuous maps?

Openness

Compactness

Boundedness only

Countability

The continuous image of a compact set is compact.

Explanation

The continuous image of a compact set is compact.

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

(0,1)

[0,1]

(0,∞)

(−∞,∞)

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

Explanation

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

9) The product of two compact sets is compact.

True

False

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

Explanation

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

10) Which is compact in ℝ?

{1/n : n∈ℕ} ∪ {0}

{1/n : n∈ℕ}

ℕ

(0,∞)

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

Explanation

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

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

Mean Value Theorem

Heine–Cantor Theorem

Bolzano–Weierstrass Theorem

Intermediate Value Theorem

Heine–Cantor ensures uniform continuity on compact sets.

Explanation

Heine–Cantor ensures uniform continuity on compact sets.

12) Compact subsets of metric spaces are always closed.

True

False

In metric spaces, compact sets are closed and bounded.

Explanation

In metric spaces, compact sets are closed and bounded.

13) Which is not compact in ℝ³?

The unit ball {‖x‖ ≤ 1}

The cube [0,1]³

ℝ³

{0}

ℝ³ is unbounded, so it cannot be compact.

Explanation

ℝ³ is unbounded, so it cannot be compact.

14) Which property is NOT guaranteed by compactness?

Existence of a convergent subsequence

Attainment of maxima and minima

Boundedness

Openness

Compactness requires closed + bounded, not openness.

Explanation

Compactness requires closed + bounded, not openness.

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

True

False

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

Explanation

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