Paths, local connectedness and their differences

Try

A path from x to y is a continuous map from [0,1] into the space with those endpoint values. In a convex subset of Euclidean space, the straight segment supplies such a path. In more complicated spaces, connectedness alone may not provide any path.

The closed topologist’s sine curve consists of the oscillating graph y=sin(1/x) for 0<x≤1 together with its limiting vertical segment at x=0. It is connected, but the vertical segment cannot be joined to the graph by a path inside the set.

Predict

Predict why adding the vertical segment makes the graph’s closure connected. The graph is a continuous image of the connected interval (0,1], and closure preserves connectedness.

Predict why this still does not supply a path from the segment to the graph. Any attempted departure from x=0 encounters oscillations of full height arbitrarily close to the departure time, conflicting with continuity of the path’s vertical coordinate.

Experiment

Separate global and local properties.

Property	Requirement
Path connected	Every pair can be joined by a path
Locally connected	Each neighborhood contains a connected open neighborhood of the point
Locally path connected	Each neighborhood contains a path-connected open neighborhood
Connected	No separation of the whole space

Local path connectedness implies local connectedness. Neither local property alone says the whole space has only one component.

Understand

Path connectedness implies connectedness because a path image is a continuous image of the connected interval. If the space had a separation, a path between opposite sides would give a connected subset meeting both sides, impossible.

Paths define equivalence classes: constant paths give reflexivity, reversing a parameter gives symmetry, and concatenation gives transitivity. These are path components. In a locally path-connected space, each path component is open: every point has a path-connected open neighborhood inside the component. The complement is a union of the other open path components, so each is also closed. A connected component cannot cross this clopen partition, while every path component is connected. Thus the two component notions agree there.

For the sine curve S, the graph is connected and its closure is exactly the graph plus the vertical segment. Every height in [−1,1] occurs along sequences with x tending to zero, while no other finite limiting points are possible. Hence S is connected.

Suppose a path joins the vertical segment to a positive-x point. Look at an interval component of the times when its x-coordinate is positive, reaching such a point. Its left endpoint has x=0. Near that endpoint, the positive continuous x-coordinate tends to zero. The intermediate value theorem forces it to attain arbitrarily small x-values with sin(1/x)=1 and −1, at times approaching the endpoint. The path’s y-coordinate therefore cannot have a limit there, contradicting continuity. This is the obstruction, not merely the visual density of the oscillations.

Apply

Use the same distinction to analyze R without zero. Each half-line is convex and path connected. No path joins the halves because its real-valued coordinate would cross zero by the intermediate value theorem. Thus the two path components are also the two connected components.

Repair “A connected drawing can always be traced continuously between any two points.” A drawing approximates the sine curve with finitely many oscillations, changing the limiting structure responsible for the counterexample.

Continue

Later proofs should name the exact connectedness property they need. Optional depth: show a space is locally connected exactly when components of every open subset are open. One direction uses connected local neighborhoods; the other uses the component of a point inside a prescribed open neighborhood.