Path Connectedness Quiz

1) Every path connected space is connected.

True

False

A path between any two points prevents the space from splitting into separate pieces.

Explanation

A path between any two points prevents the space from splitting into separate pieces.

2) The union of two path connected sets that intersect is always path connected.

True

False

Paths in each piece can be joined at the intersection.

Explanation

Paths in each piece can be joined at the intersection.

3) The set ℝ² \ {(0,0)} is path connected.

True

False

You can always go around the removed point.

Explanation

You can always go around the removed point.

4) All connected spaces are path connected.

True

False

Some spaces (e.g., topologist’s sine curve) are connected but not path connected.

Explanation

Some spaces (e.g., topologist’s sine curve) are connected but not path connected.

5) A path connected set must be convex in ℝⁿ.

True

False

Convexity is stronger than path connectedness; paths need not be straight lines.

Explanation

Convexity is stronger than path connectedness; paths need not be straight lines.

6) Which of the following subsets of ℝ² is not path connected?

A circle

Two disjoint circles

The unit disk

The line segment from (0,0) to (1,1)

Two circles are separate pieces with no path between them.

Explanation

Two circles are separate pieces with no path between them.

7) Let X = [0,1] ∪ [2,3]. Is X path connected?

Yes

No

There is a gap between 1 and 2, so no path connects points across the gap.

Explanation

There is a gap between 1 and 2, so no path connects points across the gap.

8) A set A ⊆ ℝ² is path connected if:

Any two points can be joined by a straight line

Any two points can be joined by a continuous path

The set is open

The set is compact

Path connectedness requires a continuous path, not necessarily a straight line.

Explanation

Path connectedness requires a continuous path, not necessarily a straight line.

9) Which of the following is always true?

Path connected ⇒ Connected

Connected ⇒ Path connected

Disconnected ⇒ Path connected

Path connected ⇒ Convex

Every path connected space is connected, but not every connected space is path connected.

Explanation

Every path connected space is connected, but not every connected space is path connected.

10) If a topological space X is path connected and f : X → Y is continuous, then f(X) is:

Disconnected

Path connected

Convex

Open

Continuous images of path connected sets remain path connected.

Explanation

Continuous images of path connected sets remain path connected.

11) Which of the following sets are path connected?

The interval [0,1] in ℝ

The union of two intersecting disks in ℝ²

{(x, sin(1/x)) : x ∈ (0,1]}

The set of rational numbers ℚ

A: intervals are path connected; B: intersection ensures full path connectedness; C: you can trace along the graph continuously; ℚ is totally disconnected.

Explanation

A: intervals are path connected; B: intersection ensures full path connectedness; C: you can trace along the graph continuously; ℚ is totally disconnected.

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