Singular versus Simplicial Homology of a Point; $\pi_0(X,x_0)$ versus $H_0(X,x_0)$ and $\pi_1(X,x_0)$ versus $H_1(X,x_0)$
Let $X$ be a topological space with chosen basepoint $x_0 \in X$. Then $\pi_0(X,x_0)$ is the set (groupoid) of equivalence classes of homotopy equivalence classes of maps from $(S^0, 1)$ to $X$ such that $1$ maps to $x_0$. This is just the set of path components of $X$, so that $X=\{pt\}$ gives that $\pi_0(\{pt\},\{pt\}) = 0$ [there is only one equivalence class of paths here]. Simplicial homology of a $\Delta$-complex $X$ [Hatcher] are given by $\Delta_n(X)$ the free abelian group with basis the open $n$-simplices $e_\alpha^n$ of $X$, and differential given by the boundary map \[ \partial_n(\sigma_\alpha) = \sum_i(-1)^i \sigma_\alpha|[v_0, \cdots, \hat{v_i}, \cdots, v_n] \] Viewing $X=\{pt\}$ as a $\Delta$ complex gives one $0$-cell, so that the simplicial chain complex is given by \[ 0 \rightarrow 0 \rightarrow \cdots \rightarrow 0 \rightarrow \mathbb{Z} \rightarrow 0 \cdots \] with $\mathbb{Z}$ in homological dimension $0$. Now, simplicial homology here gives \[H_i(\{pt\}) ...
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