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I'm not quite certain what Peter May had in mind 40 years ago, but probably he had in mind the fact that pushouts are a lot better behaved in CGWH than in CGH. Specifically, CGWH is closed under pushouts, one leg of which is the inclusion of a closed subspace. CGH does not have such nice behavior, and pushouts like that are used all over The Geometry of Iterated Loop Spaces, specifically in the construction of a monad from an operad and in the use of geometric realizations of simplicial spaces.