Homotopy equivalence of pairs Let $ I = [0,1] \subseteq \mathbb{R}$. I want to prove that the pair $(I^n,\partial I^n)$ is homotopy equivalent to $(\mathbb{R}^n,\mathbb{R}^n\setminus\{ 0,0,...,0 \})$, but I have a problem with the definition itself. 
Can someone please state the definition of homotopy equivalence of pairs? Apparently, I couldn't find it anywhere.

Let $A\subset X$ and $B\subset Y$ be CW-pairs (or any pairs of topological spaces such that the inclusions are cofibrations, see at the end). 
The map $f:X\to Y$ is a homotopy equivalence between the pairs $(X,A)$ and $(Y,B)$ if there is a map $g:Y\to X$ such that:


*

*$f(A)\subset B$

*$g(B)\subset A$

*There is a homotopy $H:X\times[0,1]\to X$ such that 


*

*$H(x,0)=x$ for all $x\in X$

*$H(x,1)=g\circ f(x)$ for all $x\in X$

*$H(x,t)\in A$ for all $x\in A, t\in [0,1]$


*There is a homotopy $K:Y\times[0,1]\to Y$ such that 


*

*$K(y,0)=y$ for all $y\in Y$

*$K(y,1)=f\circ g(y)$ for all $y\in Y$

*$K(y,t)\in B$ for all $y\in B, t\in [0,1]$



You can find a broader definition, where the inclusion maps $i:A\to X$ and $j:B\to Y$ are not inclusions, but only cofibrations, here as part of Proposition 1.15.
Note that not all inclusions are cofibrations, but it is the case for CW-pairs.
