What is path-connected space? — Mathematics

Definition 6.21 University Mathematics — Year 3 · Chapter 6 — General Topology

$X$ is path-connected if any two points are joined by a path (continuous $\gamma \colon [0,1] \to X$ ). Path-connected implies connected: two values of a continuous $f \colon X \to \{0,1\}$ at $x, y$ are values of the constant $f\circ\gamma$ (Theorem 6.20(1)–(2)). Convex subsets of normed spaces are path-connected (segments); so are $\R^n \setminus \{0\}$ for $n \geq 2$ (go around the origin), and $S^n$ for $n \geq 1$ (project paths from $\R^{n+1}\setminus \{0\}$ ).

The topologist’s sine curve: the graph of 1x (red) accumulates on the whole segment \0\×[-1,1] (blue). The union is connected but not path-connected: no continuous path can cross the infinitely many oscillations in finite parameter time. — The topologist’s sine curve: the graph of $\sin\frac1x$ (red) accumulates on the whole segment $\{0\}\times[-1,1]$ (blue). The union is connected but not path-connected: no continuous path can cross the infinitely many oscillations in finite parameter time.

Examples

Example 6.22 (The topologist’s sine curve)

Let $\Gamma = \{(x, \sin\frac1x) : 0 < x \leq 1\}$ and $S = \bar\Gamma = \Gamma \cup (\{0\}\times[-1,1])$ (every point $(0, y)$ , $\abs y \leq 1$ , is a limit of points of $\Gamma$ : solve $\sin\frac1x = y$ near $0$ ). Then $S$ is connected — closure of the connected $\Gamma$ , a continuous image of $(0, 1]$ (Theorem 6.20(3)) — but not path-connected: a path from $(1, \sin 1)$ to $(0,0)$ would have to traverse abscissas $\to 0$ while the ordinate oscillates between $\pm1$ ; Exercise 6.9 makes this rigorous. Connectedness and path-connectedness genuinely differ.

Read in context →

Examples

Related terms