Let $\Omega$ a smooth bounded domain in $\mathbb{R}^n$ and $\Omega_{r}^{-}=\{x\in \Omega, \rm dist(x, \partial \Omega)>r\}$
I must prove that for $r>0$ small enought, we have thar $\Omega$ and $\Omega_{r}^{-}$ are homotopically equivalent : http://mathworld.wolfram.com/HomotopyEquivalence.html
So i let $\rm f: \Omega\rightarrow \Omega_{r}^{-}$ defined by $f(x)$ such that $\rm dist(f(x),\partial\Omega)>r$ and $\rm g:\Omega_{r}^{-}\rightarrow \Omega$ defined by $g(x)$ where $\rm dist(x,\partial \Omega)>r$
after that i must prove that $f\circ g$ is homotopic to $id_{\Omega_{r}^{-}}$ and $g\circ f$ is homotopic to $id_{\Omega}$
So i define $H: \Omega_{r}^{-}\times[0,1]\rightarrow \Omega_{r}^{-}$ such that $H(x,0)=f\circ g(x), H(x,1)=id_{\Omega_{r}^{-}}(x)$
I define $H$ by $H(x,t)= t id_{\Omega_{r}^{-}}(x)+(1-t) f\circ g(x)$
But How to prove that $$\rm dist(t ~id_{\Omega_{r}^{-}}(x)+(1-t) f\circ g(x),\partial\Omega)>r$$ ?
I need your help please
Thank you