Can someone give me a simple, concrete example of a homotopy, which is not a path homotopy?
Let $f, f'$ be continuous maps from $X$ to $Y$, and let $F: X \times I\to Y$ a continuous map such that $F(x,0) = f(x)$ and $F(x, 1) = f'(x)$. Then $f$ is homotopic to $f'$.
I understand path homotopies (the path can be deformed but the endpoints stayed fixed), but now I don't see what a homotopy is. I think the notation is what is messing me up, so can you explain what are the spaces $X$, $Y$ in your example? And what are your maps $f$, $f'$?