Usually, the alleged motivation for the definition of compactly generated topological spaces is Cartesian closedness, which fails for general spaces. Of course, from a contemporary perspective, this is a perfectly reasonable explanation. Though, for those who like math history, I'd like to ask what's the unspoken history of the notion of compactly generated topological spaces, in the following sense: what were the first manifestations, technically speaking, of the failure of Cartesian closedness for general spaces? In particular, are they related to the notion of Lebesgue number of an open covering of a compact metric space?
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$\begingroup$ ncatlab.org/nlab/show/convenient+category+of+topological+spaces seems to have useful information. $\endgroup$– PatrickRMay 25 at 2:06
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$\begingroup$ There's a nice historical discussion in there! I'm trying to push it a few decades backwards by allowing it not to be explicitly spoken, as it already was in the sixties, but just 'manifest'. Maybe what I'm looking for is an example of some very early situation in which some statement involving topological spaces didn't hold due to the lack of some property that would have been guaranteed by Cartesian closedness. $\endgroup$– Dry BonesMay 25 at 3:13
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$\begingroup$ By the way, Grothendieck suggests in 'La clef des songes' a fourfold temporality of the creative process (I'll cite it in French since they are very much intelligible for English speakers, so why not): préparation, conception, travail, accomplissement. In view of this, I'd say I'm looking for the traces of Cartesian closedness during its préparation. $\endgroup$– Dry BonesMay 25 at 3:22
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