Two loops $\gamma_{1}$ and $\gamma_{2}$ with the same base point $x$ are homotopic if there exists a continuous map $H:[0,1]\times[0,1]\to X$ with the following properties: 1. $H(t,0)=\gamma_{1}(t)$ for all $t\in[0,1]$; 2. $H(t,1)=\gamma_{2}(t)$ for all $t\in[0,1]$; 3. for each $s \in[0,1]$, the map $t\mapsto H(t,s)$ is a loop with base point $x$.