I just saw the example of coffee mug and torus on the wiki page about "Homotopy" and I asked myself a few questions regarding this example:
$X=T^3$ and $Y=\mathbb{R}^3$ are topological spaces and I have $f,g: X\rightarrow Y$. My first question is: What do they mean with $f$ " takes the torus to the embedded surface-of-a-doughnut shape". So does $f$ map all points of $T^3$ to the surface of the starting doughnut? If yes wouldn't then be $f$ not continuous? Maybe one can write down $f$ explicitly and then it get's clear?
Could one also take for $X=\mathbb{R}^3$?
Many thanks in advance!