This illustrates the fact that $T\setminus\text{point}$ is homotopy equivalent to the figure eight space $E=S^1\vee S^1$. The torus $T$ can be formed by identifying edges of a square $Q$. The right hand picture shows a homotopy $h\colon [0,1]\times T\setminus\{a\}\to T\setminus\{a\}$. The left hand picture shows a compatible homotopy $k\colon [0,1]\times Q\setminus\{b\}\to Q\setminus\{b\}$.