In a paper by Rafael Dahman on the embedding of the long line into weakly complete vector spaces, those of the form $R^I$ for arbitrary $I$, and equipped with the Michal-Bastiani calculus, he notes a generalisation of Whitney's embedding theorem where he shows that the long line, equipped with a $C^r$ structure can be so embedded, but unlike Whitney, he is not able to provide a precise bound on the cardinality of $I$. He gives instead,a lower and an upper bound. However he notes that if the GCH is true then he can provide a precise cardinality.
This was news to me as I've always viewed GCH as about pure set theory - at least, this is how it has always been presented to me. To give a situation where it has geometric consequences made it seem suddenly more important. Of course, this might be - and most likely is - due to what little I know about mathematics. Hence the question:
Q. Are there geometric interpretations of GCH or/and are there geometric situations where it plays an important part?
