Path

A path in a topological space $X$ is a continuous map $\gamma\colon [0,1]\to X$, where $[0,1]$ is the interval with its usual topology. The points $\gamma(0)$ and $\gamma(1)$ are called the initial and terminal points of the path.

Paths are the basic objects used to define path-connectedness and are a special case of a curve .

Examples:

• In $\mathbb{R}^2$, for points $a,b\in\mathbb{R}^2$, the map $\gamma(t)=(1-t)a+tb$ is a path from $a$ to $b$.
• The map $\gamma(t)=(\cos(2\pi t),\sin(2\pi t))$ is a path in the unit circle with $\gamma(0)=\gamma(1)$ (a loop).