Questions: Path Connected Spaces

Question 1

Which statement correctly describes the relationship between connectedness and path-connectedness?

• A. They are equivalent properties: a space is connected if and only if it is path-connected
• B. Connected implies path-connected, but there exist path-connected spaces that are not connected
• C. Path-connected implies connected, but there exist connected spaces that are not path-connected
• D. Neither property implies the other; they are independent

Answer: (C)

Path-connected implies connected, but not conversely — this is the key asymmetry. The proof that path-connected ⟹ connected uses the fact that a continuous image of [0,1] is connected. The converse fails: the topologist's sine curve (closure of {(x, sin(1/x)) : x > 0}) is connected but not path-connected — there is no continuous path from any point on the oscillating graph to the segment {0} × [−1, 1]. Path-connectedness is the strictly stronger property.

Question 2

A mathematician claims a certain space X is connected but presents no path between two specific points. A student argues: 'If X is connected, you must be able to draw a continuous path between any two points.' Who is correct?

• A. The student — connectedness guarantees a path between any two points by definition
• B. The mathematician — connectedness does not guarantee paths; the topologist's sine curve is connected but not path-connected
• C. Both are wrong — neither property says anything about paths
• D. The student, but only for subsets of ℝⁿ

Answer: (B)

The student conflates connectedness with path-connectedness. Connectedness is a purely topological property: a space cannot be split into two disjoint nonempty open sets. It makes no direct claim about paths. Path-connectedness is the separate (stronger) condition that any two points can be joined by a continuous path. The topologist's sine curve is the standard counterexample: connected by the topological definition, yet no continuous path can reach the accumulation segment at x = 0 from the oscillating part.

Question 3

Every path-connected topological space is connected.

• T. True
• F. False

Answer: True

This is the one-directional implication that does hold. If X is path-connected, then for any two points you can find a continuous path γ: [0,1] → X between them. Since [0,1] is connected and γ is continuous, its image is connected. Using this, one can show X cannot be partitioned into two disjoint nonempty open sets, which is the definition of connectedness. The implication is: path-connected ⟹ connected.

Question 4

Nearly every connected topological space is path-connected.

• T. True
• F. False

Answer: False

False — the topologist's sine curve is the standard counterexample. It is the closure of {(x, sin(1/x)) : x > 0} in ℝ². This set cannot be split into two disjoint nonempty open sets (it is connected in the topological sense), but there is no continuous path from any point on the oscillating graph part to any point on the limiting segment {0} × [−1, 1]. The oscillation near x = 0 is too rapid for any continuous function to 'cross' it. This shows connectedness is strictly weaker than path-connectedness.

Question 5

Describe the topologist's sine curve and explain why it is connected but not path-connected.

Model answer: The topologist's sine curve is the closure of the set {(x, sin(1/x)) : x > 0} in ℝ². It consists of the oscillating graph (which oscillates infinitely rapidly as x → 0⁺) together with the limiting segment {0} × [−1, 1] on the y-axis. It is connected because any open set separating the two pieces would have to be open in ℝ², but the oscillating part accumulates at every point of the segment, preventing any separation. It is not path-connected because no continuous function γ: [0,1] → X can travel from a point on the oscillating graph to a point on the segment: near t = 0, the path would have to cross the y-axis with a continuously oscillating value, which no continuous function can do.

This example is pedagogically important because it shows that the topological definition of connectedness captures something real but weaker than geometric intuition suggests. Informally, we think 'connected' means 'you can get from here to there,' but the topological definition doesn't guarantee that. Path-connectedness is the formal version of that intuition. For the spaces of analysis and geometry — open subsets of ℝⁿ, manifolds, convex sets — the two notions coincide, which is why the distinction rarely matters in those settings.