> [!definition] > > Let $X, Y$ be [[Topological Space|topological space]] and $f_0, f_1 \in C(X, Y)$ be [[Continuity|continuous]]. > > A **homotopy** connecting $f_0$ and $f_1$ is a continuous mapping $F: X \times I \to Y$ such that $F(x, 0) = f_0(x)$ and $F(x, 1) = f_1(x)$. For each $t \in [0, 1]$, denote $f_t(x) = F(x, t)$. > > The mappings $f_0$ and $f_1$ are **homotopic** if there exists a homotopy connecting them, denoted $f_0 \simeq f_1$. The relation $\simeq$ is an [[Equivalence Relation|equivalence relation]], and the equivalence class of $f \in C(X, Y)$ is its **homotopy class**, denoted $[f]$. > > Let $A \subset X$, then the homotopy $F$ is **relative** to $A$ if $f_t|_A$ is constant with respect to $t$. The mappings $f_0$ and $f_1$ are homotopic **relative** to $A$ if there exists a homotopy $F$ between $f_0$ and $f_1$, relative to $A$.