Curve

A curve in a topological space $X$ is a continuous map $\gamma\colon I\to X$, where $I\subseteq\mathbb{R}$ is an interval (with its usual topology).

A path is a curve with domain $[0,1]$. Curves allow other parameter intervals, which is convenient for describing parametrizations and restrictions.

Examples:

• The map $\gamma(t)=(t,t^2)$ (for $t\in\mathbb{R}$) is a curve in $\mathbb{R}^2$ tracing a parabola.
• If $\gamma\colon [0,2]\to X$ is continuous, then its restriction to $[0,1]$ is a curve (and in fact a path) in $X$.