Connectedness and Path Connectedness: Concepts and Examples

A space is connected if it cannot be split into two disjoint nonempty open pieces: it is all one piece. Path connectedness is stronger: any two points join by a continuous path, a map from [0,1] into the space. Every path-connected space is connected, but the converse famously fails.

Connectedness powers the intermediate value theorem: a continuous real function on a connected space has interval-like image, attaining every intermediate value. Path connectedness is what lets robots plan motions, and the gap between the notions reveals subtleties like the topologist's sine curve.

Key Concepts

• Connected: the only clopen sets are empty set and X; no separation into two disjoint nonempty opens.
• Path connected: any x, y joined by a path gamma: [0,1] → X.
• Components: maximal connected subsets partition the space; path components refine them.
• Key implication: [0,1] is connected and continuous images of connected sets are connected, so path connected implies connected.
• Topologist's sine curve: closure of sin(1/x)'s graph for x > 0 plus the segment at x = 0: connected but not path connected.

How It Works

To prove disconnectedness, exhibit a separation. Example: the rationals Q in R's subspace topology are disconnected.

1. Let alpha = sqrt(2), irrational, and set U = Q ∩ (-∞, alpha), V = Q ∩ (alpha, ∞).
2. U, V are open in Q as intersections with R-opens; 1 ∈ U, 2 ∈ V, so both nonempty.
3. They are disjoint: no rational equals alpha; their union is all of Q.
4. Thus Q = U ∪ V is a separation: Q is disconnected. Irrationals punch uncountably many such gaps.

Common Misconceptions

• Connected ⇏ path connected: on the topologist's sine curve, sin(1/x) oscillates infinitely near 0, so no path reaches the limit segment, though the closure is one piece.
• Point deletion varies: removing a point disconnects R but leaves R^2 connected.
• Components are always closed but need not be open; in Q each component is a single closed, non-open point.

Real-World Applications

• Calculus: the intermediate value theorem is connectedness of continuous images, applied to intervals.
• Robotics: two configurations connect by motion exactly when they share a path component.
• Image processing: segmenting objects means finding connected components of pixel regions.

Connected means one piece with no open separation; path connected means any two points join by a path, which always implies connectedness. Q splits at sqrt(2) into two disjoint nonempty relative opens, proving it disconnected, while the topologist's sine curve shows the implication cannot be reversed.