---- > [!theorem] Theorem. ([[small simplices theorem]]) > Let $X$ be a [[topological space]], and $\mathcal{U}=\{ U_{\alpha} \}_{\alpha \in I}$ be a collection of [[subspace topology|subspaces]] of $X$ whose [[topological interior|interiors]] [[cover]]: $\bigcup_{\alpha \in I} \mathring{U}_{\alpha}=X.$ > Denote by $C_{n}^{\mathcal{U}}(X) \subset C_{n}(X)$ the [[submodule]] [[submodule generated by a subset|generated by]] 'small [[singular homology|singular]] [[singular simplex|simplicies]]' $\sigma:\Delta^{n} \to X$ satisfying $\sigma(\Delta^{n}) \subset U_{\alpha}$ for some $\alpha$. > > It is clear that if $\sigma$ lies in $U_{\alpha}$, then so do its faces, and therefore so does $d \sigma$. So $C_{\bullet}^{\mathcal{U}}(X)$ is a sub-[[chain complex of modules|chain complex]] of $C_{\bullet}(X)$. We denote $H_{n}^{\mathcal{U}}(X):=H_{n}(C_{\bullet}^{\mathcal{U}}(X))$. > > The *small simplices theorem* says that computing the [[singular homology]] $H_{\bullet}(X)$ of $X$ is the same as computing $H_{\bullet}^{\mathcal{U}}(X)$. Specifically, the [[homomorphism on homology induced by a chain map|natural morphism]][^1] $H_{*}^{\mathcal{U}}(X) \to H_{*}(X)$ > is an [[isomorphism]]. > [!proof]- Proof. ([[small simplices theorem]]) > Omitted in this course. ---- #### [^1]: That is, the morphism to which the [[homomorphism on homology induced by a chain map|homology functor]] sends the inclusion of $C_{\bullet}^{\mathcal{U}}(X)$ into $C_{\bullet}(X)$. ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```