Homotopy

Homotopy: a continuous deformation $f_t$ from $f = f_0$ to $g = f_1$, i.e. one map $X\times I\to Y$.

Rel $A$: points of $A$ stay put throughout.

Key tool: gluing homotopies end to end uses the pasting lemma.

Tiny example: any two maps $X\to\mathbb{R}^n$ are homotopic via $(1 - t)f + tg$.