Classifying Connected Components Quiz

1) In ℝ, every connected component must be an interval or a single point.

True

False

In ℝ, connected sets are exactly intervals (including points).

Explanation

In ℝ, connected sets are exactly intervals (including points).

2) How many connected components does X = (0,2) ∪ (3,4) have?

1

2

3

4

(0,2) and (3,4) are two components.

Explanation

(0,2) and (3,4) are two components.

3) A connected component can be a singleton.

True

False

Points are connected sets (e.g., in ℚ).

Explanation

Points are connected sets (e.g., in ℚ).

4) Classify the components of X = ℝ \{0,2}.

Two open intervals

Three open intervals

Two half-open intervals

Infinitely many isolated points

(-∞,0), (0,2), (2,∞).

Explanation

(-∞,0), (0,2), (2,∞).

5) Which sets have exactly one connected component?

ℝ

[0,1] ∪ [2,3]

Cantor set

(0,1)

ℝ and (0,1) are intervals → connected.

Explanation

ℝ and (0,1) are intervals → connected.

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6) If a component contains more than one point, it must contain an entire interval between them (in ℝ).

True

False

Connected sets in ℝ are intervals.

Explanation

Connected sets in ℝ are intervals.

7) Classify components of X = {0} ∪ (1,2) ∪ {5}.

1 component

2 components

3 components

4 components

{0}, (1,2), {5}.

Explanation

{0}, (1,2), {5}.

8) In the discrete topology on an uncountable set X, which statements are true?

Every component is a singleton

Uncountably many components

Exactly one component

All components non-closed

Each point is isolated → singleton components.

Explanation

Each point is isolated → singleton components.

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9) Components of a subspace of ℝ can be infinite in number.

True

False

Example: ℚ has infinitely many singleton components.

Explanation

Example: ℚ has infinitely many singleton components.

10) How many connected components does X = {1/n : n ≥ 1} have?

1

Countably many

Uncountably many

None

Each 1/n is isolated → countably many components.

Explanation

Each 1/n is isolated → countably many components.

11) Which sets have singleton connected components?

Cantor set

ℚ

(0,1) ∪ {2}

ℤ

Cantor set, ℚ, ℤ are totally disconnected.

Explanation

Cantor set, ℚ, ℤ are totally disconnected.

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12) The closure of every connected component is itself a connected component.

True

False

Components are maximal connected closed sets.

Explanation

Components are maximal connected closed sets.

13) How many components does X = (0,1) ∪ (1,2) ∪ (2,3] have?

1

2

3

4

Three intervals.

Explanation

Three intervals.

14) Which sets can have exactly two connected components?

(−∞,0) ∪ (0,∞)

(0,1) ∪ (2,3)

ℚ ∩ [0,1]

(0,1] ∪ {3}

Each of these splits into exactly two pieces.

Explanation

Each of these splits into exactly two pieces.

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15) Two connected components can have closures that overlap, even though the components do not intersect.

True

False

(0,1) and (1,2) closures meet at 1.

Explanation

(0,1) and (1,2) closures meet at 1.

Classifying Connected Components Quiz

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