Connected Components Reasoning Quiz

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| Questions: 15 | Updated: Dec 1, 2025
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1) If A is a connected subset of a space X, then A must lie entirely inside a single connected component of X.

Explanation

Each connected set lies inside exactly one maximal connected set.

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About This Quiz
Connected Components Reasoning Quiz - Quiz

Ready to test your topological reasoning? This quiz takes you deeper into the logic behind connected components. You’ll examine how to prove whether points lie in the same component, how to show maximal connectedness, and how separations determine component structure. You’ll also evaluate when closures intersect, how unions of connected... see moresets behave, and how to justify connectedness using contradiction. By the end, you’ll confidently apply rigorous reasoning to classify and analyze connected components in topological spaces. see less

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2) To show two points x and y lie in the same connected component, it is enough to show:

Explanation

A path or any connected set containing both places them in the same component.

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3) If the union of two connected sets intersects, then the union is connected.

Explanation

Connected sets sharing a point form a connected union.

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4) Suppose A and B are connected and A ∩ B ≠ ∅. To prove any point of A ∪ B lies in the same component as a point of A, which argument is correct?

Explanation

A ∪ B is connected, so all its points lie in the same component.

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5) Which methods can be used to prove a subset C is a connected component of X?

Explanation

A component is connected and maximal; any intersecting connected set stays inside.

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6) If every connected subset of X is a singleton, then proving components are singletons requires no further work.

Explanation

If all connected sets are singletons, components must be singletons.

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7) To show C is not a connected component, which reasoning is valid?

Explanation

If C sits in a bigger connected set, it is not maximal.

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8) Suppose X = A ∪ B, where A and B are disjoint nonempty connected sets, each closed in X. Then X has at least two components.

Explanation

Two disjoint closed connected subsets cannot join into one component.

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9) To prove the component containing x equals the union C_x = ⋃{A : A connected and x∈A}, what key step completes the proof?

Explanation

A component is characterized by maximal connectedness.

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10) Which statements justify that (0,1) is a connected component of (0,1) ∪ (2,3)?

Explanation

(0,1) cannot be enlarged while staying connected in the subspace.

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11) If two connected components have closures that intersect, then one must be contained in the closure of the other.

Explanation

Closures may touch without containment (e.g., (0,1) and (1,2)).

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12) To prove a connected component is closed, the correct reasoning is:

Explanation

Its closure is connected; being maximal, it equals its closure.

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13) Which statements show x and y cannot lie in the same component?

Explanation

Lack of any connecting connected set or a separation separates components.

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14) To prove a set C is connected, a common strategy is:

Explanation

Connectedness means there is no separation.

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15) If X is union of countably many connected subsets all sharing one point, X is connected.

Explanation

Common intersection point ensures union is connected.

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If A is a connected subset of a space X, then A must lie entirely...
To show two points x and y lie in the same connected component, it is...
If the union of two connected sets intersects, then the union is...
Suppose A and B are connected and A ∩ B ≠ ∅. To prove any point...
Which methods can be used to prove a subset C is a connected component...
If every connected subset of X is a singleton, then proving components...
To show C is not a connected component, which reasoning is valid?
Suppose X = A ∪ B, where A and B are disjoint nonempty connected...
To prove the component containing x equals the union C_x = ⋃{A : A...
Which statements justify that (0,1) is a connected component of (0,1)...
If two connected components have closures that intersect, then one...
To prove a connected component is closed, the correct reasoning is:
Which statements show x and y cannot lie in the same component?
To prove a set C is connected, a common strategy is:
If X is union of countably many connected subsets all sharing one...
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