Kerodon

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Remark 1.1.7.4 (Connected Components of $\operatorname{Sing}_{\bullet }(X)$). Let $X$ be a topological space. We let $\pi _0(X)$ denote the set of path components of $X$: that is, the quotient of $X$ by the equivalence relation

\[ (x \sim y ) \Leftrightarrow ( \exists p: [0,1] \rightarrow X) [ p(0) =x \text{ and } p(1) = y]. \]

It follows from Remark 1.1.6.23 that we have a canonical bijection $\pi _0( \operatorname{Sing}_{\bullet }(X) ) \simeq \pi _0(X)$. That is, we can identify connected components of the simplicial set $\operatorname{Sing}_{\bullet }(X)$ (in the sense of Definition 1.1.6.8) with path components of the topological space $X$.