Example 3.5.3.5. We say that a topological space $X$ is *weakly contractible* if the projection map $f: X \rightarrow \ast $ is a weak homotopy equivalence (in other words, $X$ is weakly contractible if the singular simplicial set $\operatorname{Sing}_{\bullet }(X)$ is a contractible Kan complex). Using Remark 3.5.3.4, we see that $X$ is weakly contractible if and only if it is path connected (that is, the set $\pi _0(X)$ is a singleton) and the homotopy groups $\pi _{n}(X,x)$ are trivial for $n > 0$ and any choice of base point $x \in X$ (assuming that $X$ is path connected, this condition is independent of the choice of base point).

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