I am looking for a reference or a proof of the following fact:


Let $X\_{1}\subset X\_{2}\subset\dots $ be a sequence of (hausdorff) topological spaces indexed by natural numbers such that each $X\_{i}\subset X\_{i+1}$ is a closed subspace for any $i\in \mathbb{N}$. We define $X=colim\_{i\in \mathbb{N}}X\_{i}$. 


Then the $H\_{m}(X,\mathbb{Z})=colim\_{i\in \mathbb{N}}H\_{m}(X\_{i},\mathbb{Z})$ for any natural number $m\in\mathbb{N}$. 



Directed colimit and homology
This is a Theorem on page 115 of Peter May's book [A concise course in algebraic topology](http://math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf).


A discussion can be found [here](https://math.stackexchange.com/questions/1341832/when-will-homology-and-direct-limit-commute?rq=1).


