Example 1.2.2.3. Let $X$ be a topological space and let $\operatorname{Sing}_{\bullet }(X)$ be its singular simplicial set. Then:

Vertices of $\operatorname{Sing}_{\bullet }(X)$ can be identified with points of $X$.

Edges of $\operatorname{Sing}_{\bullet }(X)$ can be identified with continuous paths $p: [0,1] \rightarrow X$. Here the source of $p$ is the point $x = p(0)$, and the target of $p$ is the point $y = p(1)$.