Disconnected Sets Problem-Solving Quiz

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| Questions: 15 | Updated: Dec 1, 2025
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1) If a set in ℝ contains two nonempty disjoint open subsets whose union is the set, it is disconnected

Explanation

This is the definition of a disconnected set.

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About This Quiz
Disconnected Sets Problem-solving Quiz - Quiz

Ready to apply connectedness reasoning to more involved problems? This quiz helps you analyze how sets behave when points are removed, intervals are separated, or subsets are combined. You’ll work with discrete sets, unions of intervals, and geometric structures in ℝ² to determine whether they break into multiple components. You’ll... see morealso examine classic misconceptions, such as whether removing a point always disconnects an interval. By the end, you’ll strengthen your ability to reason about disconnectedness using both geometric insight and topological logic. see less

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2) Every finite subset of ℝ with more than one point is connected.

Explanation

Finite sets have isolated points → disconnected.

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3) Removing a single point from a connected set always produces a disconnected set

Explanation

Removing an endpoint keeps (0,1] connected.

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4) A set consisting of a countably infinite number of isolated points is disconnected.

Explanation

Isolated points create many separate pieces.

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5) A disconnected set can have exactly one component.

Explanation

Disconnected sets must have at least two components.

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6) Which of the following sets in ℝ is disconnected?

Explanation

There is a gap between [0,1] and [2,3].

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7) Let A = {x ∈ ℝ : x ≤ -1 or x ≥ 1}. Which statement is true?

Explanation

A = (−∞,−1] ∪ [1,∞) → two separated parts.

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8) Which of these sets is connected?

Explanation

[0,1] is a single interval; all others have gaps.

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9) Let X = {x ∈ ℝ : x² < 1 or x² > 4}. Which is true?

Explanation

X = (−1,1) ∪ (−∞,−2) ∪ (2,∞).

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10) A set S ⊂ ℝ² consisting of two disjoint circles is:

Explanation

Two separate circles → disconnected.

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11) Identify all disconnected sets:

Explanation

A is finite with isolated points; C has a gap.

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12) Which of the following are disconnected?

Explanation

ℤ is disconnected; (−∞,−1] and [1,∞) have a gap.

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13) Select all disconnected subsets of ℝ²:

Explanation

Separated segments and circles are disconnected.

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14) Which sets are disconnected in ℝ?

Explanation

A and B have gaps; C is connected because 1/n → 0.

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15) Which statements are true about disconnected sets?

Explanation

C is false; D is false.

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If a set in ℝ contains two nonempty disjoint open subsets whose...
Every finite subset of ℝ with more than one point is connected.
Removing a single point from a connected set always produces a...
A set consisting of a countably infinite number of isolated points is...
A disconnected set can have exactly one component.
Which of the following sets in ℝ is disconnected?
Let A = {x ∈ ℝ : x ≤ -1 or x ≥ 1}. Which statement is true?
Which of these sets is connected?
Let X = {x ∈ ℝ : x² < 1 or x² > 4}. Which is true?
A set S ⊂ ℝ² consisting of two disjoint circles is:
Identify all disconnected sets:
Which of the following are disconnected?
Select all disconnected subsets of ℝ²:
Which sets are disconnected in ℝ?
Which statements are true about disconnected sets?
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