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Construction Let $X$ be a topological space. We define a simplicial set $\operatorname{Sing}_{\bullet }(X)$ as follows:

  • To each object $[n] \in \operatorname{{\bf \Delta }}$, we assign the set $\operatorname{Sing}_{n}(X) = \operatorname{Hom}_{\operatorname{Top}}( | \Delta ^ n |, X )$ of singular $n$-simplices in $X$.

  • To each non-decreasing map $\alpha : [m] \rightarrow [n]$, we assign the map $\operatorname{Sing}_{n}(X) \rightarrow \operatorname{Sing}_{m}(X)$ given by precomposition with the continuous map

    \[ | \Delta ^{m} | \rightarrow | \Delta ^{n} | \]
    \[ (t_0, t_1, \ldots , t_ m) \mapsto ( \sum _{\alpha (i) = 0} t_ i, \sum _{\alpha (i) = 1} t_ i, \ldots , \sum _{\alpha (i)=n} t_ i). \]

We will refer to $\operatorname{Sing}_{\bullet }(X)$ as the singular simplicial set of $X$. We view the construction $X \mapsto \operatorname{Sing}_{\bullet }(X)$ as a functor from the category of topological spaces to the category of simplicial sets, which we will denote by $\operatorname{Sing}_{\bullet }: \operatorname{Top}\rightarrow \operatorname{Set_{\Delta }}$.