My question: Why the generator of local homology group $H_{n}(\mathbb{R}^n,\mathbb{R}^n-\{0\})$ is that simplex $\sigma$, where $\sigma=[v_0,e_1,\cdots,e_n]$, $v_0=(-1,-1,\cdots,-1)\in \mathbb{R}^{n}$ $e_i=(0,0,\cdots,1,\cdots,0),i=1,\cdots ,n$.
When I was studying local topological orientation, which is defined by the local homology group. I meet an important exercise in the book: Prove that the simplex $[v_0,e_1,\cdots,e_n]$ is the generator of local homology group $H_{n}(\mathbb{R}^n,\mathbb{R}^n-\{0\})$.
I only know it is closed cycle in the homology group but don't know how to prove it is a generator. Although algebraic topology - What exactly are the elements of a local homology group? - Mathematics Stack Exchange is a little useful, for example it tells me that I can regrad the generator as a ccw or cw orientational simplex / disk. But now no answer tells me a how to prove rigorously the simplex $[v_0,e_1,\cdots,e_n]$ is the generator of local homology group $H_{n}(\mathbb{R}^n,\mathbb{R}^n-\{0\})$.
Look forward to your answer sincerely. Thanks for your reading.