what is the 0'th homology group of $\mathbb{Q}$ I mean $ H_{0}(\mathbb{Q})$?as the 0'th homology group is counting the path component of the space so it should be infinite direct sum of copies of $\mathbb{Z}$,and that exactly the place I feel doubt and I asked myself is it true for infinitely points?
so please help me and guide me ,thank you very much.
From the definition of singular homology, the $n$th homology group of $\mathbb{Q}$ is trivial except for $H_0(\mathbb{Q};\mathbb{Z})$ which is countably infinitely freely generated abelian, that is $H_0(\mathbb{Q};\mathbb{Z})\cong\displaystyle\bigoplus_{n=1}^{\infty}\mathbb{Z}$.
(really the only cases where you can directly apply the definition of singular homology is when your space is totally disconnected).