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In Milnor's Construction of Universal Bundles, II, he defines $E_nG$ by repeated joins of $G$ with itself, but he has to use the `strong topology' on the join instead of the everyday topology that results from viewing the join as the pushout of $X\times CY \gets X\times Y\to CX \times Y$. He makes use of the strong topology in verifying that the orbit map $E_nG \to B_nG$ is a bundle.

My question is: if we restrict our attention to weak Hausdorff compactly generated spaces, is the strong topology needed?

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    $\begingroup$ @Jeff: This is not an answer but kind of echoes the problem for joins. In my book now available as "Topology and Groupoids" (amazon, see my web page) I gave an account of the initial topology on joins, and it is convenient for mah purposes. For example it is associative; it is good for maps into it. The identification type topology is of course good for maps out of it. But I never managed to prove they were equivalent in compactly generated spaces. Am I missing something simple? $\endgroup$
    – Ronnie Brown
    Commented Mar 5, 2013 at 21:11

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