Path Connectedness Problem-Solving Quiz

1) Let A = {(x,y) ∈ ℝ² : y = x²}. Which reasoning correctly proves A is path connected?

It is bounded.

It is the image of a continuous function t ↦ (t,t²)

It is symmetric about the y-axis.

It contains the origin.

A = image of ℝ under continuous map t → (t,t²), so A is path connected.

Explanation

A = image of ℝ under continuous map t → (t,t²), so A is path connected.

2) Suppose X ⊆ ℝ² is such that any two points can be joined by a piecewise-linear path inside X. Then:

X is arc connected

X is path connected

X is convex

X must be open

A polygonal path is a continuous curve → arc connected → path connected.

Explanation

A polygonal path is a continuous curve → arc connected → path connected.

3) Let B = ℝ² ∖ {(0,y): y ≤ 0}. Why is B path connected?

Removing a point never breaks path connectedness

Removing a ray in the plane does not disconnect it

The set is convex

It is homeomorphic to a closed disk

A downward half-ray does not separate the plane, so paths can go around it.

Explanation

A downward half-ray does not separate the plane, so paths can go around it.

4) Which of the following sets is not path connected?

ℝ ∖ {(1,2,3)}

ℝ² ∖ {(0,0)}

{(x,y): xy = 0}

{(x,y): x² + y² = 1}

The union of axes disconnects into pieces not path connected.

Explanation

The union of axes disconnects into pieces not path connected.

5) Let C = {(x,y): x>0} ∪ {(x,y): y>0}. Why is this set path connected?

Both sets are convex and overlap in the first quadrant

They intersect only at the axes

Each point has a unique path to the origin

It is the union of two disjoint open sets

Both half-planes meet in the first quadrant → one connected region.

Explanation

Both half-planes meet in the first quadrant → one connected region.

6) Which approaches can correctly prove a set S is path connected?

Show any two points can be connected by a broken line

Parameterize the set as a continuous image

Show S is the union of connected sets

Construct an explicit continuous map between any two points

Broken lines, parameterization, and explicit paths prove path connectedness; union alone insufficient.

Explanation

Broken lines, parameterization, and explicit paths prove path connectedness; union alone insufficient.

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7) Consider D = {(x,y): 0 < x ≤ 1, 0 < y ≤ 1/x}. Which statements justify that D is path connected?

Each vertical slice is an interval

The region is star-shaped from (1,1)

It is convex

Paths can stay under y=1/x

Vertical slices are intervals; continuous curves staying under 1/x connect points.

Explanation

Vertical slices are intervals; continuous curves staying under 1/x connect points.

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8) Let E = {(r cosθ, r sinθ): 1 < r < 2, 0 ≤ θ < 2π}. Valid reasons this set is path connected:

It is an annulus

All points can be connected by radial lines

Radial path may exit annulus

It is homotopy equivalent to a circle

Annulus is path connected; homotopy equivalence preserves path connectedness.

Explanation

Annulus is path connected; homotopy equivalence preserves path connectedness.

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9) Which sets below are path connected?

ℝ ∖ {(1,2,3)}

ℝ² ∖ {(0,0)}

{(x,y): xy > 0}

{(x,y): y = sin(1/x), x>0}

Plane minus point is path connected; continuous curve is path connected.

Explanation

Plane minus point is path connected; continuous curve is path connected.

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10) Which strategies can prove path connectedness for S = {(x,y): y ≥ |x|}?

Show the set is convex

Construct straight-line paths

Use piecewise paths above |x|

Angle-based parametrization

Region is convex; straight or piecewise paths remain inside.

Explanation

Region is convex; straight or piecewise paths remain inside.

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11) Which facts imply plane minus finite points is path connected?

Paths can go around missing points

Finite removal doesn't disconnect plane

Each removed point creates isolated component

Polygonal paths avoid removed points

ℝ² minus finite points stays connected; polygonal paths avoid them.

Explanation

ℝ² minus finite points stays connected; polygonal paths avoid them.

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12) Which statements are necessarily true for path connected sets?

It is connected

It is the image of a continuous function

It is arc connected

Any two points can be joined

Path connected ⇒ connected and any two points have a path.

Explanation

Path connected ⇒ connected and any two points have a path.

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13) Show T = {(x,y,z): x²+y² = 1, 0 ≤ z ≤ 5} is path connected. Valid reasons:

It is a cylinder

Product of path connected spaces

It is convex

It is bounded

Circle × interval is path connected.

Explanation

Circle × interval is path connected.

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14) Which of the following sets is path connected, and why?

Two touching circles

Two tangent circles

Two separated circles

One inside another, disjoint

Intersection point joins paths.

Explanation

Intersection point joins paths.

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15) Let H = {(x,y): y = eˣ}. Which arguments prove path connected?

Image t↦(t,eᵗ)

Graph of continuous function

It is convex

None

Continuous curve → path connected; not convex.

Explanation

Continuous curve → path connected; not convex.

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Path Connectedness Problem-Solving Quiz

1) Let A = {(x,y) ∈ ℝ² : y = x²}. Which reasoning correctly proves A is path connected?

2)

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

2) Suppose X ⊆ ℝ² is such that any two points can be joined by a piecewise-linear path inside X. Then:

3) Let B = ℝ² ∖ {(0,y): y ≤ 0}. Why is B path connected?

4) Which of the following sets is not path connected?

5) Let C = {(x,y): x>0} ∪ {(x,y): y>0}. Why is this set path connected?

6) Which approaches can correctly prove a set S is path connected?

7) Consider D = {(x,y): 0 < x ≤ 1, 0 < y ≤ 1/x}. Which statements justify that D is path connected?

8) Let E = {(r cosθ, r sinθ): 1 < r < 2, 0 ≤ θ < 2π}. Valid reasons this set is path connected:

9) Which sets below are path connected?

10) Which strategies can prove path connectedness for S = {(x,y): y ≥ |x|}?

11) Which facts imply plane minus finite points is path connected?

12) Which statements are necessarily true for path connected sets?

13) Show T = {(x,y,z): x²+y² = 1, 0 ≤ z ≤ 5} is path connected. Valid reasons:

14) Which of the following sets is path connected, and why?

15) Let H = {(x,y): y = eˣ}. Which arguments prove path connected?