Theorem 41.1:

Let $X$ be a topological space and $\{X_{\alpha}\;/\; \alpha \in \Lambda\}$ be a directed system of open subsets of $X$ such that every compact subset of $X$ lies in some $X_{\alpha}$. For a pair of indices $\alpha \leq \beta$, the map $f_{\alpha\beta}:H_n(X_{\alpha})\longrightarrow H_n(X_{\beta})$ is the homomorphism induced by inclusion $X_{\alpha}\longrightarrow X_{\beta}$. Then, the family $\{H_n(X_{\alpha})\;/\alpha\in \Lambda\}$ together with the maps $f_{\alpha\beta}$ forms an inductive system of abelian groups and

$$\varinjlim_{\alpha}H_n(X_{\alpha}) = H_n(X) = H_n\big(\varinjlim_{\alpha}X_{\alpha}\big) \eqno(41.1)$$