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Remark Recall that a topological space $X$ is contractible if the projection map $X \rightarrow \ast $ is a homotopy equivalence. Equivalently, $X$ is contractible if the identity map $\operatorname{id}_{X}: X \rightarrow X$ is homotopic to the constant function $X \rightarrow \{ x\} \hookrightarrow X$, for some base point $x \in X$. It follows from Example that every contractible topological space is weakly contractible. In particular, for each $n \geq 0$, the standard simplex $| \Delta ^ n |$ is weakly contractible.