I have heard many mentions of Milnor's construction of the classifying space of a topological group, but I am only now learning the details. The first surprising piece that I had to learn was that I was using the wrong topology on the join: the Milnor construction requires the initial topology, which is distinct from the usual description of joins as quotients. Luckily, I have found some appropriate references: I am using §5.7 of Brown's *[Topology and Groupoids][1]* to become comfortable with this topology, and §14.4 of Dieck's *Algebraic Topology* to understand contractibility. 

While Dieck's shifting argument makes sense to me, I am confused about the proliferation of a different argument, which I have often seen elsewhere. Let $E_n=G*G*\cdots$ be the $n$-fold join for any $n=1,2,\dots,\infty$, with the initial topology. Then apparently: "each inclusion $E_n\hookrightarrow E_{n+1}$ is null-homotopic and thus the colimit $E_\infty$ is contractible." There are two pieces of this that I have yet to understand:

- Why is $E_n\hookrightarrow E_{n+1}$ always a cofibration? (This is needed for the colimit to be contractible.) I haven't thought about this too much, because I've been preoccupied with my second question. So perhaps this part is easier.

- Is it indeed true that $\text{colim}\,E_n\cong E_\infty$? I see that the identity map $\text{colim}\,E_n\hookrightarrow  E_\infty$ is continuous, but don't understand why its inverse should be. After all, I've been told to be distrusting of limits and colimits commuting!

Are these things true and can this argument actually be used for a general $G$? Or does this only work in special cases (perhaps $G$ being compact Hausdorff)? Any help is much appreciated!

  [1]: https://groupoids.org.uk/pdffiles/topgrpds-e.pdf