Disconnected Sets Quiz

1) A subset (A ⊂ R) is disconnected if:

It is empty

It can be written as the union of two nonempty disjoint open sets in the subspace topology

It contains all its limit points

It is bounded

Disconnected means split into two nonempty, disjoint open subsets.

Explanation

Disconnected means split into two nonempty, disjoint open subsets.

Ready to understand how and why sets break apart? This quiz introduces you to disconnected sets—sets that can be separated into two nonempty, disjoint open subsets in the subspace topology. You’ll analyze examples in ℝ and ℝ², determine which sets split into multiple components, and explore how disconnectedness behaves unde... see moreunions, complements, and closures. Through these problems, you’ll build a strong intuition for how sets become separated and why intervals remain connected. By the end, you’ll confidently identify disconnected sets and explain the structural properties that cause them to fall apart.
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2)

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2) Which set is disconnected in R?

[0,1]

[0,1] ∪ [2,3]

(0,1)

Gap.

Explanation

Gap.

3) If X is disconnected, which true?

X not connected

X has exactly two pts

X compact

X open

Definition.

Explanation

Definition.

4) Which interval not disconnected?

[0,1] ∪ [2,3]

(0,2)\{1}

[0,3]

(1,2)∪(2,3)

[0,3] interval.

Explanation

[0,3] interval.

5) Empty set is:

Connected

Disconnected

Neither

Both

Convention.

Explanation

Convention.

6) A={x∈R: x^2≤1 or x^2≥4}:

Connected

Disconnected

Empty

Compact and connected

Multiple pieces.

Explanation

Multiple pieces.

7) X⊂R^n disconnected if exist nonempty U,V:

U∩V=∅ & U∪V=X

U∩V≠∅

U=V

U open, V closed

Definition.

Explanation

Definition.

8) Select disconnected:

[0,1]

[0,1] ∪ [2,3]

{1,2,3}

(0,1)

Separated or finite >1.

Explanation

Separated or finite >1.

Submit

9) Which true for disconn sets?

Must be bounded

Can be finite

Have ≥2 components

Removing any pt disconnects connected set

True.

Explanation

True.

Submit

10) Disconnected examples:

[0,1]∪[3,4]

{0,1,2}

(0,2)\{1}

[0,2]

Gaps.

Explanation

Gaps.

Submit

11) If X disconnected, operations produce disconnected:

Union w another disconnected

Intersection

Closure

Complement

Complement splits.

Explanation

Complement splits.

12) Disconnected in R2:

Two horiz segments

Unit circle

Open disk

Axes union

A,D disconnect.

Explanation

A,D disconnect.

Submit

13) X disconnected if:

2 pts not path-join

∃ continuous surjection to {0,1}

Finite

Bounded

Charact.

Explanation

Charact.

Submit

14) True for disconn X:

≥2 clopen subsets

Exactly 2 components

Every subset disconn

∃ nonconstant map to {0,1}

Properties.

Explanation

Properties.

Submit

15) True about disconn sets:

Two isolated pts disconn

Union disjoint connected always disconn

Remove any pt disconnects

Finite >1 disconn

Correct.

Explanation

Correct.

Submit