04 - Paths

Path

Let $U\subseteq {\mathbb{R}}^2$ and $a_0,a_1\in U$

A path in $U$ between $a_0$ and $a_1$ is a continuous map $\gamma$ defined by

$$\gamma :[0,1]\to U\text{ with }\gamma (0)=a_1\text{ and }\gamma (1)=a_1$$

Homotopic

Let $U\subseteq {\mathbb{R}}^2$ and $a_0,a_1\in U$
Let ${\gamma }_0,{\gamma }_1$ be paths in $U$ connecting $a_0,a_1$

Then paths ${\gamma }_0,{\gamma }_1$ are homotopic if there exists $\Gamma$ defined by

$$\Gamma :[0,1]\times [0,1]\to U$$

with

$$\begin{array}{rlrlrlrl}& \Gamma (s,0)=a_0& & \text{ and }& & \Gamma (s,1)=a_1& & \text{ for all }s\in [0,1]\\ & \Gamma (0,t)={\gamma }_0(t)& & \text{ and }& & \Gamma (1,t)={\gamma }_1(t)& & \text{ for all }t\in [0,1]\end{array}$$

Can be useful to treat $\Gamma$ as a way of interpolation between the two paths