[Math] Homology functor commute with direct limit

Question

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 $h$?

Answer 1

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.