Definition. Let $A$ be a closed subspace of a topological space $X$. We say that $(X, A)$ is a good pair if there is a neighborhood $V$ of $A$ in $X$ which deformation retracts to $A$.


In the proof of Proposition 2.22 in Hatcher's *Algebraic Topology*, the author shows the following:


Fact. Let $(X, A)$ be a good pair and $V$ be a neighborhood of $A$ in $X$ which deformation retracts to $A$. Then the inclusion map $i: (X, A)\to (X, V)$ induces an isomorphism $i_*:H_n(X, A)\to H_n(X, V)$ for all $n$.


The proof given in the book is a one liner. We just use the long exact sequence for the triple $(X, V, A)$ along with the fact that $H_n(V, A)$ is $0$ for all $n$.


But I have no intuition for the above solution and as of now the use of the long exact sequence seems like an algebraic trick to me. So I tried a more direct approach.


Claim. $i_*$ is injective.
Proof. Let $\sigma\in C_n(X)$ be arbotrary with $\partial \sigma\in C_{n-1}(A)$ and suppose that $i_*([\bar \sigma])=0$ (We use $\bar \sigma$ to denote the coset of $\sigma$ in $C_n(X, A)$ and $[\bar \sigma]$ to denote the homology class of $\bar \sigma$). So $\bar \sigma$ is a boundary in $Z_n(X, V)$. This means that there is $\tau\in C_{n+1}(X)$ such that $\bar \sigma=\partial\bar \tau$ in $C_n(X, V)$ (the meaning of the 'bars' have been suitably changed). So $\sigma -\partial \tau\in C_n(V)$. Note that $\partial (\sigma -\partial \tau)\in C_{n-1}(A)$, which means that $\overline{\sigma-\partial \tau}\in Z_n(V, A)$. But since $H_n(V, A)=0$, we have $Z_n(V, A)=B_n(V, A)$. Thus $\overline{\sigma-\partial \tau}=\partial \bar\theta$ for some $\theta\in C_{n+1}(V)$. This leads to $\sigma-\partial(\tau+\theta)\in C_n(A)$, giving $\bar \sigma=\partial\overline{(\tau+\theta)}$ in $C_n(X, A)$. This shows that $[\bar \sigma]=0$ in $H_n(X, A)$ and we are done.


The Problem. I am having trouble showing the surjectivity of $i_*$.


Can somebody please help me with this and also, if possible, throw some light on the use of the long exact sequence.


Thanks.