Path Connectedness Quiz

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| Questions: 15 | Updated: Dec 1, 2025
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1) Every path connected space is connected.

Explanation

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

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About This Quiz
Path Connectedness Quiz - Quiz

Ready to understand one of the most intuitive forms of connectedness? This quiz introduces you to path connectedness—the idea that any two points in a space can be joined by a continuous path. You’ll examine examples in ℝ and ℝ², compare path connectedness with connectedness, and explore when unions of... see moresets remain path connected. Through these problems, you’ll build intuition for how paths behave in topological spaces and why intersections matter. By the end, you’ll be able to identify path connected sets, understand when path connectedness fails, and apply the concept to geometric and abstract spaces. see less

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

Explanation

Paths in each piece can be joined at the intersection.

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

Explanation

You can always go around the removed point.

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4) All connected spaces are path connected.

Explanation

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

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

Explanation

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

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

Explanation

Two circles are separate pieces with no path between them.

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

Explanation

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

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8) A set A ⊆ ℝ² is path connected if:

Explanation

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

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9) Which of the following is always true?

Explanation

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

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

Explanation

Continuous images of path connected sets remain path connected.

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11) Which of the following sets are path connected?

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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12) A space X is path connected if:

Explanation

This is the definition of path connectedness.

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13) Let A = {(x, y) : x² + y² = 1 or x² + y² = 4}. Which are true?

Explanation

Each circle is path connected, but the two circles are separate, so A is disconnected.

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14) Which statements are correct regarding path connectedness?

Explanation

Continuous images of path connected sets are path connected, and the product of two path connected spaces is path connected.

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15) Consider the topologist’s sine curve S = {(x, sin(1/x)) : x ∈ (0,1]} ∪ {(0, y) : y ∈ [-1,1]}. Which are correct?

Explanation

S is connected but is a classic example of a space that is not path connected.

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Every path connected space is connected.
The union of two path connected sets that intersect is always path...
The set ℝ² \ {(0,0)} is path connected.
All connected spaces are path connected.
A path connected set must be convex in ℝⁿ.
Which of the following subsets of ℝ² is not path connected?
Let X = [0,1] ∪ [2,3]. Is X path connected?
A set A ⊆ ℝ² is path connected if:
Which of the following is always true?
If a topological space X is path connected and f : X → Y is...
Which of the following sets are path connected?
A space X is path connected if:
Let A = {(x, y) : x² + y² = 1 or x² + y² = 4}. Which are true?
Which statements are correct regarding path connectedness?
Consider the topologist’s sine curve S = {(x, sin(1/x)) : x ∈...
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