Connected Components Reasoning Quiz

1) If A is a connected subset of a space X, then A must lie entirely inside a single connected component of X.

True

False

Each connected set lies inside exactly one maximal connected set.

Explanation

Each connected set lies inside exactly one maximal connected set.

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

There exists a continuous path between them.

They are in the same open set.

They lie in some connected subset of X.

Their closures intersect.

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

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.

True

False

Connected sets sharing a point form a connected union.

Explanation

Connected sets sharing a point form a connected union.

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?

Components are closed.

Use that A ∪ B is connected.

Every point of B is a limit point of A.

Components are open.

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

Explanation

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

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

Prove C is connected.

Prove any connected set intersecting C lies in C.

Prove C is open and closed.

Prove C is maximal among connected subsets.

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

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.

True

False

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

Explanation

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

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

Show its closure is connected.

Show a larger connected set contains C.

Show C contains more than one point.

Show C is closed.

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

Explanation

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

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.

True

False

Two disjoint closed connected subsets cannot join into one component.

Explanation

Two disjoint closed connected subsets cannot join into one component.

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

Show Cₓ is closed.

Show any connected set intersecting Cₓ is inside it.

Show Cₓ is maximal connected.

Show Cₓ is open.

A component is characterized by maximal connectedness.

Explanation

A component is characterized by maximal connectedness.

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

No connected set contains points in both intervals.

(0,1) is maximal connected in the space.

(0,1) is open and closed in the subspace.

Any connected subset intersecting (0,1) stays in it.

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

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.

True

False

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

Explanation

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

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

Closure of a connected set is connected; maximality forces equality.

Complement of a component is open.

Components are always open.

Connected sets must be closed.

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

Explanation

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

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

Every connected set containing x excludes y.

There exists a separation placing x and y in different pieces.

Closures of their components are disjoint.

There is no path between x and y.

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

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:

Show it is compact.

Show every continuous function is bounded.

Show any supposed separation leads to contradiction.

Show it contains limit points.

Connectedness means there is no separation.

Explanation

Connectedness means there is no separation.

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

True

False

Common intersection point ensures union is connected.

Explanation

Common intersection point ensures union is connected.

Connected Components Reasoning Quiz

1) If A is a connected subset of a space X, then A must lie entirely inside a single connected component of X.

2)

What first name or nickname would you like us to use?

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

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

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?

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

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

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

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.

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

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

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

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

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

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

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