Given the space $X=\mathbb{R}$, how can we calculate its first homology group $H_1(\mathbb{R})$? Intuitively, the object of first homology describes 1-dimensioal holes in the set which here doesn't exist, hence maybe $$H_1(\mathbb{R})=\{0\}?$$ If my intuition is right: how do we prove that?
EDIT: I'm using simplicial homology means the quotient group $$H_n(X)=\ker\partial_n\big/\text{Im}\partial_{n+1}$$
Since you are using simplicial homology, you first need to equip $\Bbb R$ with a $\Delta$-complex structure. You can do so by choosing the integers $\tau_n:\Delta^0\to\{n\}$ as the $0$-simplices, and the closed intervals $\sigma_n:\Delta^1\to[n,n+1]$ as the $1$-simplices. Then $\partial_1:C_1(X)\to C_0(X)$ sends $\sigma_n$ to $\tau_{n+1}-\tau_n$. Can you see why $\text{Ker}(\partial_1)$ is trivial?