If we have $X=S^2$ and $A$ is a finite set of $k$ points in $X$, I would like to show that $\tilde{H}_0(A)\simeq\mathbb{Z}^{k-1}$ and $\tilde{H}_i(A)=0$ for $i>0$.
I understand the definition of relative homology but I am still new to it, I find it hard to find some clue on how to proceed.
Here is my attempt:
We have the long exact sequence: $\ldots\to\tilde{H}_0(A)\xrightarrow{i_*}\tilde{H_0}(X)\xrightarrow{j_*}\tilde{H}_0(X/A)\to0$.
I guess $H_0(A)=\mathbb{Z}^k$. But how can we relate to $\tilde{H}_0(A)$? Could somebody please give some hints and guidance on how to proceed?
I have been looking at many sources for the definition of long exact sequence, relative homology, but still find it quite hard to apply the concepts in solving problems.
Any helps are greatly appreciated. Thanks!