Relative homotopy

Remarks

For $A = \emptyset$ this is an ordinary homotopy. Being homotopic relative to $A$ is an equivalence relation on the continuous maps $X \to Y$ that agree with a given map on $A$. Example: for $X = [0, 1]$, $A = \set{0, 1}$, $Y = \R^2$, and two paths $f, g$ with $f(0) = g(0)$ and $f(1) = g(1)$, the map $H(s, t) = (1 - t) f(s) + t g(s)$ is a homotopy relative to $A$, since $H(0, t) = f(0)$ and $H(1, t) = f(1)$. Non-example: in $\R^2 \setminus \set{0}$, the upper and lower half circles from $(1, 0)$ to $(-1, 0)$ are homotopic as maps, since $[0, 1]$ is contractible, but they are not homotopic relative to $\set{0, 1}$.