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In "Construction of Universal Bundles, II", Milnor has to replace the standard topology on the join with what he calls the "strong topology" which is the smallest topology such that certain maps are continuous. The purpose is to force the obvious local sections of his orbit map to be continuous, so that the orbit map will be a fiber bundle.

Essentially the same trick is used in later work of Hall on the generalized Whitney sum and in Strøm's "The Homotopy Category is a Homotopy Category."

My question: Is Milnor's paper the origin of this trick?

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    $\begingroup$ @Jeff, This won't help probably, but for completeness, I've seen a similar game in the paper of Dold and Lashof: Principal quasi-fibrations and fibre homotopy equivalence of bundles. Illinois J. Math. 3 1959 285–305. They say they pattern their construction after Milnor's paper. $\endgroup$
    – John Klein
    Commented Jan 16, 2011 at 17:20

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This is not an answer to the question, but a reference to a more recent appearance of such a trick. The teardrop topology is defined for a map $p:X \to Y \times {\mathbb R}$ to be the smallest topology on the disjoint union $X \sqcup Y$ such that the inclusion $X \subseteq X \sqcup Y$ is an open embedding and the collapse map $c:X \sqcup Y \to Y \times (-\infty,\infty]$ is continuous. This is used in the 2000 Topology paper Neighborhoods in stratified spaces with two strata by Hughes, Taylor, Weinberger and Williams.

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