Piecewise linear homotopy

Definition

Suppose $f,g:X\to Y$ are continuous maps with $Y$ a subset of a (possibly infinite-dimensional) Euclidean space with the subspace topology. Suppose there exist continuous maps $f_{0},f_{1},f_{2},\dots ,f_{n}:X\to Y$ such that $f=f_{0}$ and $g=f_{n}$ and linear homotopies $F_{i(i+1)}$ between each $f_{i}$ and $f_{i+1}$. Then, we can define a composite of homotopies $F_{i(i+1)}$ which is a homotopy from $f$ to $g$. A homotopy obtained in this way is termed a piecewise linear homotopy.