Proof:

The fact that $\{H_n(X_{\alpha})\;/\alpha\in \Lambda\}$ is an inductive system is clear. Let $A$ denote the inductive limit of this system in AbGr and $f_{\alpha}:H_n(X_{\alpha})\longrightarrow A$ denote the associated homomorphisms described in definition (40.2). The inclusion maps $X_{\alpha}\subset X$ induce homomorphisms $\iota_{\alpha}:H_n(X_{\alpha})\longrightarrow H_n(X)$. To simplify notations, we shall suppress the bar and use the same symbol $\zeta$ to denote a cycle as well as the homology class it represents. The proof of (41.1) hinges on two simple facts:

(i)

If $\zeta^{\prime}$ is an $n-$chain in $X$ then there exists an $\alpha \in \Lambda$ such that the images of the constituent simplicies in $\zeta^{\prime}$ are all contained in $X_{\alpha}$. We shall say that the chain $\zeta^{\prime}$ is supported in $X_{\alpha}$. Thus $\zeta^{\prime}$ may be viewed as a singular chain in $X_{\alpha}$ and the latter will be provisionally denoted by $\zeta$ in the proof. Further if $\zeta^{\prime}$ is a cycle in $X$ then $\zeta$ is a cycle in $X_{\alpha}$ and $\zeta^{\prime} = \iota_{\alpha}(\zeta).$

(ii)

If $\zeta^{\prime}$ is a boundary of a chain $\omega^{\prime}$ in $X$ then there exists a $\beta \in \Lambda$ such that $\zeta^{\prime}$ and $\omega^{\prime}$ are both supported in $X_{\beta}$ and the relation $\zeta = \partial \omega$ holds in $X_{\beta}$. In other words,

$$\iota_{\alpha}(\zeta) = 0\;\;$$ implies $$\;\;f_{\alpha\beta}(\zeta) = 0\;\;$$ for some $$\;\;\beta \geq \alpha. \eqno(41.2)$$

To prove these note that the image of each singular simplex is a compact subset of $X$ and each chain is a finite linear combination of singular simplicies.

Property (2) of definition (40.2) may now be applied to the family of homomorphisms $\iota_{\alpha}$. There exists a group homomorphism $\phi : A\longrightarrow H_n(X)$ such that

$$\phi\circ f_{\alpha} = \iota_{\alpha},\quad \alpha \in \Lambda \eqno(41.3)$$

To show that $\phi$ is surjective, by (i) above, an arbitrary cycle $\zeta^{\prime}$ in $X$ with support in $X_{\alpha}$ representing an element of $H_n(X)$ may be expressed as $\iota_{\alpha}({\zeta})$ where $\zeta$ is a cycle in $X_{\alpha}$. By (41.3) we see that ${\zeta}^{\prime} \in$   im$\;\phi.$ To show that $\phi$ is injective, let $\zeta^{\prime} \in A$ be such that $\phi({\zeta}^{\prime}) = 0$ in $X$. By exercise 4 of lecture 40, we can write

$$\zeta^{\prime} = \sum f_{\alpha}(\zeta_{\alpha}) \eqno(41.4)$$

where the sum is finite and each $\zeta_{\alpha}$ is a cycle in $X_{\alpha}$. Choose a $\beta$ exceeding all the indices in (41.4) and for each index $\alpha$ in (41.4), $f_{\alpha}(\zeta_{\alpha}) = f_{\beta}\circ f_{\alpha\beta}(\zeta_{\alpha})$ and so using (41.3),

$$0 = \phi(\zeta^{\prime}) = (\phi\circ f_{\beta})\big(\sum f_{\alp... ...{\alpha})\big) = \iota_{\beta}\big(\sum f_{\alpha\beta}(\zeta_{\alpha})\big)$$

Invoking (41.2) we arrive at $\sum f_{\alpha\beta}(\zeta_{\alpha}) = 0$ (perhaps with a larger $\beta$). Applying $f_{\beta}$ we see that $\zeta^{\prime} = 0$ as desired. $\square$