I'm trying to understand a homotopy between two path.
the definition of path in Topological space $X$ from point $x$ to $y$ is a continues map $f:[0,1]\rightarrow X$ such that $f(0)=x$ and $f(1)=y$.
A homotopy between to paths in Topological space $X$ from point $x$ to $y$;
$f:[0,1]\rightarrow X$ and $g:[0,1]\rightarrow X$ is a continuous map $H:[0,1] \times [0,1]\rightarrow X$ such that $H(0,x)=f(x)$, $H(1,x)=g(x)$, $H(t,0)=x$ and $H(t,1)=y$.
How can I define the homotopy of paths with general interval like $[0,a]$ ?
Assume we have two paths in Topological space $X$ from point $x$ to $y$;
$f:[0,a]\rightarrow X$ and $g:[0,b]\rightarrow X$ what is the neutral way to define ?
Best Answer
In general for any topological spaces $X,Y$ and continuous maps $f,g:X\to Y$, a homotopy from $f$ to $g$ is a continuous map $H:X\times[0,1] \to Y$ such that $H(x,0)=f(x)$ and $H(x,1)=g(x)$ for all $x\in X$.
This generalizes your definition of a homotopy between paths. To do it for a general interval you can let $X=[a,b]$, but this is really the same definition as doing it for $X=[0,1]$ by just choosing a homeomorphism $h:[0,1]\to[a,b]$ and reparametrizing.
Note however that homotopies only make sense between maps who has the same domain and codomain. So a homotopy from a map $f:[0,a]\to X$ to a map $g:[0,b]\to X$ doesn't really make sense since they don't have the same domain. However, if you reparametrize such that the domains become the same, you can make sense of a homotopy between them. In other words, it might be useful to choose homeomorphisms $h_f:[0,1]\to [0,a]$, $h_g:[0,1] \to [0,b]$ and consider homotopies between $f\circ h_f$ and $g \circ h_g$.