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Let $(X_i,\phi_i^j)$ be a directed system of topology spaces and its direct limit is $(X,\phi_i)$

$$\lim_{\rightarrow}(X_i,\phi_i^j)=(X,\phi_i)$$

Since $H_n$ ($n^{th}\, homology \,\,group$ ) is functor so $(H_n(X_i),(\phi_i^j)_*)$ is a directed system in category of abelian groups such that $(\phi_j)_*(\phi_i^j)_*=(\phi_i)_*$ for every $i\leq j$.

I know direct limit exist for any directed system in category of groups then we can assume

$$\lim_{\rightarrow}(H_n(X_i),\phi_{i^j_*})=(G,f_i)$$

and by defination of direct limit there exit a unique homomorphism

$$h:G\rightarrow H_n(X)$$ such that $\phi_{i_*}=h(f_i)$ for every i.

If I show $h$ is an isomorphism then

$$H_n(\lim_{\rightarrow}(X_i,\phi_i^j))\cong\lim_{\rightarrow}\left(H_*(X_i),(\phi_i)_*\right)$$

Can some body help me to proving the bijection of $h$?

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It is not true that $h$ is an isomorphism in general. For instance let $X=S^1$ and consider the directed system of all countable subspaces of $X$ (with their inclusion maps). Then $X$ is the direct limit of this system (since a subset of $X$ is closed iff it is sequentially closed). But $H_1(X_i)=0$ for each $i$ in the system, since each $X_i$ is totally disconnected. So $$\lim_{\rightarrow}\left(H_1(X_i),(\phi_i)_*\right)=0$$ whereas $$H_1(\lim_{\rightarrow}(X_i,\phi_i^j))=\mathbb{Z}$$ for this system.

The directed system of topoloical spaces gives a directed sistem of complexes $S_i = S(X_i)$ where $S$ is the singular chains functor. However, it is not necessarily true that $\varinjlim S_i = S$. If this were the case, the following general considerations apply.

Consider any diagram $I$, and the functor $\varinjlim : A^I \to A$. This is right exact, and then there is a canonical map

$$\theta : \varinjlim H(S_i) \to H(\varinjlim S_i)$$

which may fail to be an isomorphism. However, if your indexing set $I$ is *directed*, then the functor $\varinjlim : A^I \to A$ is *exact*, and exact “functors commute with homology”. More precisely, there is a map

$$\psi : H(\varinjlim S_i)\to \varinjlim H(S_i)$$

which is an inverse to $\psi$.

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