Path Connectedness Proof Strategies Quiz

1) To prove a space X is path connected, it is enough to show that for any two points x, y ∈ X there exists a continuous map f : [0,1] → X with f(0)=x and f(1)=y.

True

False

That is the definition of path connectedness.

Explanation

That is the definition of path connectedness.

2) To prove a subset of ℝ² is path connected, showing a polygonal (piecewise-linear) path exists between any two points is valid.

True

False

A polygonal line is a continuous path, so it works.

Explanation

A polygonal line is a continuous path, so it works.

3) If a set is the image of a connected interval under a continuous map, then proving path connectedness is automatic.

True

False

Continuous images of intervals are connected, but not always path connected.

Explanation

Continuous images of intervals are connected, but not always path connected.

4) When proving a union A ∪ B is path connected, it is always enough to show that both A and B are path connected.

True

False

They must also intersect to join their paths.

Explanation

They must also intersect to join their paths.

5) To prove that a continuous map f : X → Y yields a path connected image, it is sufficient to show that the image of every path in X is a path in Y.

True

False

Continuous images of paths remain paths.

Explanation

Continuous images of paths remain paths.

6) Which method below is a valid proof strategy for showing a set in ℝ² is path connected?

Show the set is convex

Show every open cover has a finite subcover

Show the set is dense in a connected set

Show the interior is non-empty

Convex sets always allow straight-line paths.

Explanation

Convex sets always allow straight-line paths.

7) To prove that a graph consisting of two edges sharing exactly one common vertex is path connected, we should:

Show each edge is path connected, and they intersect

Show each edge is open

Show each edge is closed

Show each edge is bounded

Two connected pieces with a common point form a path connected union.

Explanation

Two connected pieces with a common point form a path connected union.

8) Let A = {(x,0) : 0 ≤ x ≤ 2} ∪ {(1,y) : 0 ≤ y ≤ 2}. To prove A is path connected, the most appropriate method is:

Exhibit a polygonal path between arbitrary points

Show that the set is convex

Show that both the horizontal and vertical segments are compact

Show the projection onto the x-axis is connected

We can connect any two points using broken-line paths through the corner (1,0).

Explanation

We can connect any two points using broken-line paths through the corner (1,0).

9) You want to prove a subset S ⊆ ℝⁿ is path connected by showing it is the union of infinitely many path connected subsets. Which additional property is needed?

Each subset must be open

The subsets must pairwise intersect

The collection must be nested or chain-connected

The union must be compact

They need overlap in a chain so a path can move between pieces.

Explanation

They need overlap in a chain so a path can move between pieces.

10) To prove the open annulus A = {(x,y) : 1 < x² + y² < 4} is path connected, the simplest approach is:

Show every point can be connected to a reference point by a radial path

Show the annulus is the union of two disjoint circles

Show the annulus is disconnected but not totally disconnected

Show it is compact

You can always move radially then around the circle to link points.

Explanation

You can always move radially then around the circle to link points.

11) Which of the following are valid methods for proving a set is path connected?

Showing the set is star-shaped

Showing the set is the continuous image of a path connected set

Showing any two points can be joined by a continuous curve

Showing the boundary of the set is connected

Star-shaped, continuous images, and explicit curves all guarantee path connectedness.

Explanation

Star-shaped, continuous images, and explicit curves all guarantee path connectedness.

Submit

12) Which of the following strategies will always succeed in proving path connectedness?

Displaying a specific path between any two arbitrary points

Showing the set is arc connected

Showing the set is path connected at one point

Showing the set is connected and locally path connected

A and B are direct proofs; D ensures global path connectedness.

Explanation

A and B are direct proofs; D ensures global path connectedness.

Submit

13) To prove that S = {(x,y) : y = x³, x ∈ ℝ} is path connected, one may:

Define a path f(t) = ((1−t)x₁ + t x₂, (1−t)x₁ + t x₂)³

Show the set is the image of ℝ under a continuous function

Show the graph is increasing

Parametrize the curve using t ↦ (t, t³)

S is the continuous image of ℝ via t ↦ (t,t³), so it is path connected.

Explanation

S is the continuous image of ℝ via t ↦ (t,t³), so it is path connected.

Submit

14) You want to prove this set is path connected: T = {(x,y) : x>0, y>0}. Which methods are valid?

Observe it is convex

Construct the path f(t)=((1−t)x₁ + t x₂, (1−t)y₁ + t y₂)

Show it is a dense subset of a convex set

Show it is homeomorphic to (0,∞)²

The first quadrant is convex, straight-line paths stay inside, and it is homeomorphic to a rectangle.

Explanation

The first quadrant is convex, straight-line paths stay inside, and it is homeomorphic to a rectangle.

Submit

15) Let H = {(x,y) : xy = 1}. To prove H is path connected, one may:

Parametrize the hyperbola as t ↦ (t, 1/t)

Show it has two disconnected branches

Show each branch is path connected

Demonstrate no path can cross the axes

Each branch is path connected via t → (t,1/t). But the two branches do not connect, so H is not globally path connected.

Explanation

Each branch is path connected via t → (t,1/t). But the two branches do not connect, so H is not globally path connected.

Submit

Path Connectedness Proof Strategies Quiz

1) To prove a space X is path connected, it is enough to show that for any two points x, y ∈ X there exists a continuous map f : [0,1] → X with f(0)=x and f(1)=y.

2)

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

2) To prove a subset of ℝ² is path connected, showing a polygonal (piecewise-linear) path exists between any two points is valid.

3) If a set is the image of a connected interval under a continuous map, then proving path connectedness is automatic.

4) When proving a union A ∪ B is path connected, it is always enough to show that both A and B are path connected.

5) To prove that a continuous map f : X → Y yields a path connected image, it is sufficient to show that the image of every path in X is a path in Y.

6) Which method below is a valid proof strategy for showing a set in ℝ² is path connected?

7) To prove that a graph consisting of two edges sharing exactly one common vertex is path connected, we should:

8) Let A = {(x,0) : 0 ≤ x ≤ 2} ∪ {(1,y) : 0 ≤ y ≤ 2}. To prove A is path connected, the most appropriate method is:

9) You want to prove a subset S ⊆ ℝⁿ is path connected by showing it is the union of infinitely many path connected subsets. Which additional property is needed?

10) To prove the open annulus A = {(x,y) : 1 < x² + y² < 4} is path connected, the simplest approach is:

11) Which of the following are valid methods for proving a set is path connected?

12) Which of the following strategies will always succeed in proving path connectedness?

13) To prove that S = {(x,y) : y = x³, x ∈ ℝ} is path connected, one may:

14) You want to prove this set is path connected: T = {(x,y) : x>0, y>0}. Which methods are valid?

15) Let H = {(x,y) : xy = 1}. To prove H is path connected, one may: