Remark 3.4.6.2. In the situation of Theorem 3.4.6.1, the canonical maps $\operatorname{Sing}_{\bullet }(U) \hookleftarrow \operatorname{Sing}_{\bullet }(U \cap V) \hookrightarrow \operatorname{Sing}_{\bullet }(V)$ are monomorphisms. Consequently, the conclusion of Theorem 3.4.6.1 is equivalent to the assertion that the natural map
\[ \operatorname{Sing}_{\bullet }(U) \coprod _{ \operatorname{Sing}_{\bullet }( U \cap V) } \operatorname{Sing}_{\bullet }(V) \rightarrow \operatorname{Sing}_{\bullet }(X) \]
is a weak homotopy equivalence of simplicial sets (see Proposition 3.4.2.11).