Singular simplicial set

Summary

The singular simplicial set $\operatorname{Sing} X$ of a topological space $X$ is a simplicial set. Its $n$-simplices are the continuous maps from the standard topological $n$-simplex into $X$. It turns a space into combinatorial data without losing its weak homotopy type. Its chains give singular homology, and it is a Kan complex, so its simplicial homotopy groups are the homotopy groups of $X$.