Connected Components Quiz

1) Every connected component of a topological space is a connected subset.

True

False

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

Explanation

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

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

True

False

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

Explanation

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

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

A smallest open connected set containing a point

A maximal connected subset of X

Any path-connected subset of X

The closure of a connected set

Connected components are maximal connected subsets.

Explanation

Connected components are maximal connected subsets.

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

1

2

3

Infinitely many

X splits into (-∞,0) and (0,∞).

Explanation

X splits into (-∞,0) and (0,∞).

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

True

False

Connected components are closed in the full space.

Explanation

Connected components are closed in the full space.

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

A single point

The entire set

Any finite subset

Any open subset

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

Explanation

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

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

All singleton sets

The entire set X

All open sets

All closed sets

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

Explanation

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

8) Connected components always partition the space.

True

False

Every point belongs to exactly one component.

Explanation

Every point belongs to exactly one component.

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

[0,1]

ℚ with the usual topology

ℝ

(0,1)\{2}

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

Explanation

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

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

Open intervals

Singletons

Countable sets

None of the above

Total disconnectedness means all components are points.

Explanation

Total disconnectedness means all components are points.

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

True

False

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

Explanation

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

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

1

2

3

4

Each interval and isolated point forms its own component.

Explanation

Each interval and isolated point forms its own component.

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

They are maximal connected subsets.

They are always open.

They form a partition of the space.

They are always closed.

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

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:

The set of points reachable from x by a path

The smallest closed connected set containing x

The union of all connected subsets containing x

The intersection of all connected subsets containing x

Component = union of all connected sets containing x.

Explanation

Component = union of all connected sets containing x.

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

True

False

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

Explanation

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

Connected Components Quiz

1) Every connected component of a topological space is a connected subset.

2)

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

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

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

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

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

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

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

8) Connected components always partition the space.

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

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

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

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

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

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

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