Path Connectedness

Real Analysis · Axiom Academy

A stronger kind of connectedness: any two points can be joined by a continuous path that never leaves the set. 1. Joining Two Points Without Leaving A path in a set S is a continuous map . It starts at and ends at , and crucially every intermediate point also lies in S — the walk never steps outside. Sets you can always walk across Any convex set (a disk, a ball, an interval, all of ): the straight segment stays inside, so it is automatically path connected. The sphere for , and any region you can draw without lifting your pen. First recall: S is connected if it cannot be written as with U,V disjoint, non-empty, and open. The main theorem says the path version is the stronger one. 3. The Topologist's Sine Curve Does connected go back the other way? No. Here is the standard counterexample — connected, but with no path from one part to the other. 4. No Path Reaches the Segment S is connected, yet you cannot walk from the segment V to the oscillating curve. The oscillation is the obstruction. The gap between the two notions only appears in pathological sets like the sine curve. For an open subset of , connected and path connected mean exactly the same thing. You can now tell connectedness from the stronger path connectedness, and you know exactly where they part ways. Scroll up to revisit any step.

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