Path

Let $(X,d)$ be a metric space. A path in $X$ is a continuous function

for some real interval $[a,b]\subset\mathbb{R}$ (often $[0,1]$). The endpoints of the path are $\gamma(a)$ and $\gamma(b)$.

Paths model “continuous motion” inside a space. Path-connectedness is stronger than connectedness and is often easier to verify in geometric settings.

Examples:

• In $\mathbb{R}^k$, the map $\gamma(t)=(1-t)x+ty$ for $t\in[0,1]$ is a path from $x$ to $y$ (a line segment).
• On the unit circle $S^1\subset\mathbb{R}^2$, $\gamma(t)=(\cos t,\sin t)$ for $t\in[0,2\pi]$ is a path with $\gamma(0)=\gamma(2\pi)$.
• In the discrete metric space $X$ (with at least two points), any continuous map from a connected interval $[a,b]$ must be constant, so there are no nontrivial paths connecting distinct points.