$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Theorem 3.4.6.1 (Excision). Let $X$ be a topological space, and let $U,V \subseteq X$ be subsets whose interiors $\mathring {U} \subseteq U$ and $\mathring {V} \subseteq V$ comprise an open covering of $X$. Then the diagram of singular simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Sing}_{\bullet }(U \cap V) \ar [r] \ar [d] & \operatorname{Sing}_{\bullet }(U) \ar [d] \\ \operatorname{Sing}_{\bullet }(V) \ar [r] & \operatorname{Sing}_{\bullet }(X) } \]
is a homotopy pushout square (Definition 3.4.2.1).
Proof of Theorem 3.4.6.1 from Theorem 3.4.6.6.
Let $X$ be a topological space and let $\operatorname{\mathcal{U}}= \{ U, V \} $ be a pair of subsets of $X$. Then $\operatorname{Sing}_{\bullet }^{\operatorname{\mathcal{U}}}(X)$ can be identified with the pushout
\[ \operatorname{Sing}_{\bullet }(U) \coprod _{ \operatorname{Sing}_{\bullet }( U \cap V) } \operatorname{Sing}_{\bullet }(V), \]
formed in the category of simplicial sets. Theorem 3.4.6.6 then asserts that if $X = \mathring {U} \cup \mathring {V}$, then the inclusion
\[ \operatorname{Sing}_{\bullet }(U) \coprod _{ \operatorname{Sing}_{\bullet }( U \cap V) } \operatorname{Sing}_{\bullet }(V) \hookrightarrow \operatorname{Sing}_{\bullet }(X) \]
is a weak homotopy equivalence. By virtue of Remark 3.4.6.2, this is equivalent to Theorem 3.4.6.1.
$\square$