---- > [!definition] Definition. ([[path homotopy]]) > Two [[parameterized curve|paths]] $f$ and $f'$, mapping the [[closed interval|interval]] $I=[0,1]$ into $X$, are called **path-homotopic** if they have the same initial point $x_{0}$ and the same final point $x_{1}$, and if there is a [[continuous]] map $F:I \times I \to X$ s.t. $\begin{align} F(s,0) & = f(s) \text{ and } F(s,1) = f'(s), \\ F(0,t) & = x_{0} \ \ \ \text{ and } F(1,t) = x_{1} \end{align}$ for each $s \in I$ and each $t \in I$. The first condition says that $F$ is a [[homotopy]] between $f$ and $f'$, and the second says that for each $t$, the [[parameterized curve]] $f_{t}$ defined by the equation $f_{t}(s)=F(s,t)$ is a [[parameterized curve]] from $x_{0}$ to $x_{1}$. We write $f \simeq_{p} f'$. ![[CleanShot 2024-03-22 at [email protected]]] In other words, a path homotopy is precisely a [[homotopy relative to a subset|homotopy relative to]] the endpoints $x_{0},x_{1}$. > [!intuition] > Said differently, the first condition says that $F$ represents a [[continuous]] way of deforming the [[parameterized curve|path]] $f$ to the [[parameterized curve|path]] $f'$, and the second condition says that the endpoints of the [[parameterized curve|path]] remain fixed during the deformation. > [!basicexample] > Let $X$ denote the punctured plane $\mathbb{R}^{2}- \{ \b 0 \}$. The following paths in $X$, $\begin{align} f(s)= & (\cos \pi s, \sin \pi s), \\ g(s) = & (\cos \pi s, 2 \sin \pi s), \end{align}$ are [[path homotopy|path homotopic]], as witnessed by the [[straight-line homotopy]]. > > [!basicnonexample] > But the [[straight-line homotopy]] between $f$ and the [[parameterized curve]] $h(s)=(\cos \pi s, -\sin \pi s)$ > is not acceptable, for its image does not lie in $X$. [[TODO]] picture from munkres. This is intuitive, and it turns out that *no* [[path homotopy]] exists between $f$ and $h$. But this is not so easy to prove... ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```