Connected Components Quiz

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Quizzes Created: 7387 | Total Attempts: 9,527,684
| Questions: 15 | Updated: Dec 1, 2025
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1) Every connected component of a topological space is a connected subset.

Explanation

By definition, a connected component is a connected maximal subset.

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About This Quiz
Connected Components Quiz - Quiz

Ready to understand how connectedness breaks into pieces? This quiz focuses on connected components—the maximal connected subsets of a topological space. You’ll examine how components behave in ℝ, ℝ², discrete and indiscrete spaces, and totally disconnected sets like ℚ. Through these problems, you’ll learn why components partition the space, when... see morethey are closed, and how many components a set can have. By the end, you’ll be able to classify connected components, identify their structure, and explain why they are fundamental building blocks in topology. see less

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2) Two different connected components can overlap in at least one point.

Explanation

If two components share a point, they merge into one component.

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3) Which of the following best describes a connected component of a space X?

Explanation

Connected components are maximal connected subsets.

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4) Let X = ℝ \{0}. How many connected components does X have?

Explanation

X splits into (-∞,0) and (0,∞).

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5) Every connected component is always a closed subset of the space.

Explanation

Connected components are closed in the full space.

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6) In the discrete topology on a nonempty set X, each connected component is:

Explanation

In the discrete topology, every point is isolated → only singletons are connected.

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7) In the indiscrete topology on X, the connected components are:

Explanation

The only nonempty open set is X itself → the whole space is connected.

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8) Connected components always partition the space.

Explanation

Every point belongs to exactly one component.

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9) Which of the following spaces has infinitely many connected components?

Explanation

ℚ is totally disconnected → components are singletons → infinitely many.

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10) If a space is totally disconnected, its connected components must be:

Explanation

Total disconnectedness means all components are points.

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11) The union of two connected components is always connected.

Explanation

Components are maximal connected sets; their union is disconnected unless same component.

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12) Let X = (0,1) ∪ (2,3) ∪ {5}. How many connected components does X have?

Explanation

Each interval and isolated point forms its own component.

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13) Which of the following are properties of connected components?

Explanation

Components are maximal, partition the space, and are closed; not always open.

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14) The connected component containing a point x in a space X is:

Explanation

Component = union of all connected sets containing x.

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15) A connected component may fail to be open in the subspace topology.

Explanation

Connected components need not be open (e.g., components in ℚ).

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Every connected component of a topological space is a connected...
Two different connected components can overlap in at least one point.
Which of the following best describes a connected component of a space...
Let X = ℝ \{0}. How many connected components does X have?
Every connected component is always a closed subset of the space.
In the discrete topology on a nonempty set X, each connected component...
In the indiscrete topology on X, the connected components are:
Connected components always partition the space.
Which of the following spaces has infinitely many connected...
If a space is totally disconnected, its connected components must be:
The union of two connected components is always connected.
Let X = (0,1) ∪ (2,3) ∪ {5}. How many connected components does X...
Which of the following are properties of connected components?
The connected component containing a point x in a space X is:
A connected component may fail to be open in the subspace topology.
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