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?