$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Theorem (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

Proof of Theorem from Theorem 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 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, this is equivalent to Theorem $\square$