Connected Sets Quiz

1) Singleton sets in any topological space are connected.

True

False

One-point sets cannot be separated.

Explanation

One-point sets cannot be separated.

Ready to build a solid understanding of connectedness? This quiz introduces you to what it means for a set to be connected—that it cannot be split into two nonempty, disjoint open sets. You’ll explore connectedness through intervals in ℝ, unions of sets, and behavior in general topological spaces. Through these... see morequestions, you’ll learn how connected sets behave, how intersections influence connectedness, and why intervals are the fundamental connected sets in ℝ. By the end, you’ll be ready to identify connected sets and explain why certain sets break apart into disconnected pieces. see less

2)

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2) Which property is necessary for a set to be connected?

Closed

Bounded

Cannot be split into two nonempty disjoint open sets

Open

Definition of connectedness.

Explanation

Definition of connectedness.

3) The set (−∞,0) ∪ (0,∞) in ℝ is:

Connected

Not connected

Open but connected

Closed

There is a gap at 0.

Explanation

There is a gap at 0.

4) Which statement is true?

Connected sets are closed

Every interval in ℝ is connected

Union of any connected sets is connected

Intersection never connected

Only intervals result in connectedness.

Explanation

Only intervals result in connectedness.

5) When is A∪B connected?

A,B disjoint

A∩B ≠ ∅

A or B open

A or B closed

Intersection ensures connection.

Explanation

Intersection ensures connection.

6) Examples of connected sets:

(−2,1]

[0,1] ∪ [2,3]

[−1,1]

{5}

Intervals and singletons.

Explanation

Intervals and singletons.

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7) Every interval in ℝ is connected.

True

False

Intervals have no gaps.

Explanation

Intervals have no gaps.

8) Which set is connected in ℝ?

[0,1] ∪ [2,3]

(0,1) ∪ (1,2)

[0,2]

{1,3}

Only [0,2] is an interval.

Explanation

Only [0,2] is an interval.

9) A set in ℝ is connected if it cannot be expressed as the union of two nonempty, disjoint open sets.

True

False

This is the definition of connectedness.

Explanation

This is the definition of connectedness.

10) A connected set in ℝ is always an interval.

True

False

Connected subsets of ℝ are intervals.

Explanation

Connected subsets of ℝ are intervals.

11) The empty set is considered connected.

True

False

∅ has no separation.

Explanation

∅ has no separation.

12) Let A=[0,1], B=[1,2]. Then A∪B is:

Connected

Not connected

Only connected if open

Only connected if closed

They touch at 1 → [0,2].

Explanation

They touch at 1 → [0,2].

13) Which sets are connected?

[0,5)

[1,2] ∪ [3,4]

(−3,3)

{0,1}

Intervals only.

Explanation

Intervals only.

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14) Which statements are correct?

Connectedness is topological

Connected⇒compact

Open intervals connected

All multi-point sets disconnected

Connectedness is topological; open intervals connected.

Explanation

Connectedness is topological; open intervals connected.

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15) The union of two connected sets is always connected.

True

False

Only guaranteed if they intersect.

Explanation

Only guaranteed if they intersect.