Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 3.5.3.6. 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 3.5.3.3 that every contractible topological space is weakly contractible. In particular, for each $n \geq 0$, the standard simplex $| \Delta ^ n |$ is weakly contractible.