Path Connectedness Proof Strategies Quiz

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| Questions: 15 | Updated: Dec 1, 2025
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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.

Explanation

That is the definition of path connectedness.

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About This Quiz
Path Connectedness Proof Strategies Quiz - Quiz

Think you can prove a set is path connected? This quiz challenges you to use formal strategies such as constructing explicit paths, using convexity, leveraging intersections, and applying continuous images of intervals. You’ll examine unions of path connected sets, parameterizations of curves, and geometric reasoning in ℝ². These problems help... see moreyou understand which proof methods always work, which require extra conditions, and which fail. By the end, you’ll confidently apply rigorous strategies to show when a space is path connected. see less

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

Explanation

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

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

Explanation

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

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

Explanation

They must also intersect to join their paths.

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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.

Explanation

Continuous images of paths remain paths.

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

Explanation

Convex sets always allow straight-line paths.

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

Explanation

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

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

Explanation

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

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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?

Explanation

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

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

Explanation

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

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

Explanation

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

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

Explanation

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

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

Explanation

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

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

Explanation

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

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

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.

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To prove a space X is path connected, it is enough to show that for...
To prove a subset of ℝ² is path connected, showing a polygonal...
If a set is the image of a connected interval under a continuous map,...
When proving a union A ∪ B is path connected, it is always enough to...
To prove that a continuous map f : X → Y yields a path connected...
Which method below is a valid proof strategy for showing a set in...
To prove that a graph consisting of two edges sharing exactly one...
Let A = {(x,0) : 0 ≤ x ≤ 2} ∪ {(1,y) : 0 ≤ y ≤ 2}. To prove...
You want to prove a subset S ⊆ ℝⁿ is path connected by showing...
To prove the open annulus A = {(x,y) : 1 < x² + y² < 4} is...
Which of the following are valid methods for proving a set is path...
Which of the following strategies will always succeed in proving path...
To prove that S = {(x,y) : y = x³, x ∈ ℝ} is path connected, one...
You want to prove this set is path connected: T = {(x,y) : x>0,...
Let H = {(x,y) : xy = 1}. To prove H is path connected, one may:
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