Thanks for the responses. Yes, Baire category and Lebesgue measure are the two main kinds of largeness measure I was thinking of. (Yet another is Hausdorff dimension.) --Dan Andy wrote: << RWG wrote: <<<< I wrote: <<<<<< By the way, rwg, your last two sentences are quite interesting! I wonder how small, in some appropriate sense, that uncountable set S = {x in [0,1]^n | #(finv(x)) > 1} can be.

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Rather huge. For the 2D Peano-Hilbert squarefiller, S = KxC U CxK, where C := [0,1] and K := the dyadic rationals therein, with the triply visited being the dense set KxK.

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Well, that's huge in the sense that it's uncountable and dense, but it's "small" in at least two senses: It has measure 0, and it's a "meagre" set; that is, a countable union of nowhere dense sets.

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