The topologist's sine curve — the closure of the graph of sin(1/x) for x > 0 in ℝ² — is connected but not path-connected. What prevents a continuous path from reaching the segment {0} × [−1, 1]?
AThe set is not compact, so paths in it need not have closed images
Bsin(1/x) oscillates infinitely rapidly as x → 0, so no continuous function can approach the y-axis without 'jumping'
CThe segment {0} × [−1, 1] is open in the subspace topology, making it unreachable
DPaths in ℝ² cannot cross the y-axis because it divides the plane into two components
As x → 0⁺, sin(1/x) oscillates between −1 and 1 with increasing frequency and no limit. A continuous path γ: [0,1] → X approaching the y-axis segment would require traversing arbitrarily large oscillations in a finite parameter interval — impossible for a continuous function. By the intermediate value theorem, the path would need to hit every value between −1 and 1 infinitely often in any neighborhood of its endpoint. The space is connected because you cannot separate it with disjoint open sets, but the path-connectedness construction fails due to this geometric obstruction.
Question 2 Multiple Choice
You want to study the fundamental group of a space X by analyzing loops based at a point x₀. Why is path-connectedness a prerequisite rather than a convenience?
APath-connectedness is equivalent to simply-connectedness, which is the actual hypothesis needed
BWithout path-connectedness, there may be points between which no path exists, making the fundamental group depend on basepoint in a way that cannot be compared across the space
CThe fundamental group is only defined for metric spaces, which must be path-connected
DPath-connectedness ensures the space has no holes, guaranteeing a trivial fundamental group
The fundamental group π₁(X, x₀) is defined as homotopy classes of loops at x₀. For this to reflect the global topology of X, groups at different basepoints must be comparable — which requires a path between any two points (the isomorphism is constructed by conjugating loops with a connecting path). If X is not path-connected, π₁(X, x₀) only sees the path-component of x₀, and basepoints in different components give unrelated groups. Path-connectedness ensures the fundamental group is a property of the whole space, not an accident of basepoint choice.
Question 3 True / False
Every path-connected topological space is connected.
TTrue
FFalse
Answer: True
Proof sketch: Suppose X is path-connected but not connected. Then X = U ∪ V with U, V disjoint, nonempty, and open. Take p ∈ U and q ∈ V. By path-connectedness, there is a continuous path γ: [0,1] → X with γ(0) = p and γ(1) = q. Then [0,1] = γ⁻¹(U) ∪ γ⁻¹(V) is a partition into disjoint open sets, with 0 ∈ γ⁻¹(U) and 1 ∈ γ⁻¹(V). But [0,1] is connected — contradiction. Therefore path-connectedness implies connectedness.
Question 4 True / False
Most connected topological space is path-connected.
TTrue
FFalse
Answer: False
The topologist's sine curve (closure of {(x, sin(1/x)) : x > 0}) is the standard counterexample. It is connected — it cannot be split into two disjoint nonempty open sets — but it is not path-connected because no continuous path can reach the limit segment {0} × [−1, 1] from the oscillating part. Connectedness is a strictly weaker condition: it rules out global separations, but does not guarantee that any two points can be joined by a continuous arc.
Question 5 Short Answer
What is the difference between connectedness and path-connectedness, and why does algebraic topology require the stronger condition rather than just connectedness?
Think about your answer, then reveal below.
Model answer: Connectedness says only that the space cannot be split into two disjoint nonempty open sets — it is a global separation condition. Path-connectedness says any two points can be joined by a continuous path γ: [0,1] → X — it is a constructive condition. Algebraic topology requires path-connectedness because homotopy theory is built from paths: the fundamental group consists of homotopy classes of loops, and comparing fundamental groups at different basepoints requires a path between them. A merely connected space may have pairs of points with no path between them, making loop-based constructions undefined or incoherent. Path-connectedness guarantees the algebraic structure actually reflects the topology of the whole space.
The topologist's sine curve illustrates the gap: it is connected but the limit segment is completely unreachable by paths, so any homotopy theory based at a point on the oscillating part would be blind to the y-axis segment. Path-connectedness eliminates this kind of pathological disconnection between the geometry and the algebra.