Applying Compactness in Problem Solving

1) A continuous function on a compact set is always bounded.

True

False

Extreme value theorem: continuous functions on compact sets are bounded.

Explanation

Extreme value theorem: continuous functions on compact sets are bounded.

2) Which function achieves both maximum and minimum on [0,2π]?

F(x)=tan(x)

F(x)=cos(x)

F(x)=1/x

F(x)=ln(x)

cos(x) is continuous and bounded on [0,2π].

Explanation

cos(x) is continuous and bounded on [0,2π].

3) If K is compact and f:K→ℝ is continuous, then f(K) is:

Always compact

Always open

Always infinite

Always unbounded

Continuous images of compact sets are compact.

Explanation

Continuous images of compact sets are compact.

4) Which of the following subsets of ℝ² is compact?

{(x,y) : x²+y² ≤ 1}

{(x,y) : x²+y² < 1}

{(x,y) : x ≥ 0}

ℝ²

The closed unit disk is closed and bounded, hence compact.

Explanation

The closed unit disk is closed and bounded, hence compact.

5) Compactness implies sequential compactness in metric spaces.

True

False

In metric spaces, compact ⇔ sequentially compact.

Explanation

In metric spaces, compact ⇔ sequentially compact.

6) A compact set in ℝ is always:

Finite

Countable

Closed and bounded

Open

By Heine–Borel theorem.

Explanation

By Heine–Borel theorem.

7) Which theorem ensures uniform continuity of continuous functions on compact sets?

Intermediate Value Theorem

Heine–Cantor Theorem

Bolzano–Weierstrass Theorem

Fundamental Theorem of Calculus

Heine–Cantor: continuous functions on compact sets are uniformly continuous.

Explanation

Heine–Cantor: continuous functions on compact sets are uniformly continuous.

8) Which is compact in ℝ³?

The cube [0,1]³

The open unit ball {‖x‖<1}

Half-space {x>0}

ℝ³

[0,1]³ is closed and bounded, so compact.

Explanation

[0,1]³ is closed and bounded, so compact.

9) The product of finitely many compact sets is compact.

True

False

Finite products of compact sets remain compact (special case of Tychonoff).

Explanation

Finite products of compact sets remain compact (special case of Tychonoff).

10) Which of these is NOT compact in ℝ?

[0,1]

{0}

(0,1)

[2,3]

Open intervals are not compact.

Explanation

Open intervals are not compact.

11) If a sequence lies in a compact set, then:

It always converges

It has a convergent subsequence

It must be monotone

It diverges to infinity

Compactness guarantees subsequential convergence.

Explanation

Compactness guarantees subsequential convergence.

12) Which is a property of compact sets in general topological spaces?

Every open cover has a finite subcover

Every sequence converges

The set is bounded

The set is countable

Open cover definition is general.

Explanation

Open cover definition is general.

13) Compact sets are preserved under continuous functions.

True

False

Continuous images of compact sets remain compact.

Explanation

Continuous images of compact sets remain compact.

14) Which is a compact set in ℝ?

[0,1] ∪ [2,2.5]

(0,1) ∪ [2,2.5]

(−∞,∞)

(0,∞)

Union of finitely many closed and bounded sets is compact.

Explanation

Union of finitely many closed and bounded sets is compact.

15) Which property does compactness NOT guarantee?

Boundedness

Closedness

Uniform continuity of continuous maps

Differentiability

Compactness relates to boundedness and continuity, not differentiability.

Explanation

Compactness relates to boundedness and continuity, not differentiability.