In example 2.18 page 118 of Hatcher's Algebraic topology we read :
Applying the long exact sequence of reduced homology groups to a
pair $(X,x_0)$ with $x_0\in X$ yields isomorphisms $H_n(X,x_0)=\tilde H_n
(X)$ for all $n$ since $\tilde H_n (x_0)=0$ for all $n$.
Actually the long exact sequence gives that $\tilde H_n(X,x_0)=\tilde H_n
(X)$ for all $n$, i think it remains to show that $\tilde H_n(X,x_0)=H_n(X,x_0)$ for all $n$ which is equivalent to show that $\tilde H_0(X,x_0)=H_0(X,x_0)$ but how to do it? and is it true more generally that $\tilde H_0(X,A)=H_0(X,A)$ for all non empty $A$?
Best Answer
Right before Example 2.17 on the same page, Hatcher says