Warning 3.4.6.3. In the situation of Theorem 3.4.6.1, it is generally not true that the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Sing}_{\bullet }(U \cap V) \ar [r] \ar [d] & \operatorname{Sing}_{\bullet }(U) \ar [d] \\ \operatorname{Sing}_{\bullet }(V) \ar [r] & \operatorname{Sing}_{\bullet }(X) } \]
is a pushout square of simplicial sets. Concretely, this is because the image of a continuous function $f: | \Delta ^{n} | \rightarrow X$ need not be contained in either $U$ or $V$.