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Very interesting answer Thank you very much. I guess in definition of $A_2$ you mean $D_2$ . the contractibility then comms from the contractibility of logarithm branch.
@PietroMajer An amazing point: A telepathy: I was thinking to the same question and assigning a natural number $n$ as the maximum valuse of the number of partition of a continum: Then I thought that re your example in comment possibly can be reconstructed to 3 partition. If we can write $Q$ as disjoint union of two dense subset...and apply it to $\mathbb{Q}\cap [0,1]$ but I realized this idea does not work since we have a connected interval at the ground
Ok Thank you. BTW is your argument capable of being a proof of contractibility. Any way the infinite dimensional sphere is amazing: two complemented dense contractible set!
Thank you for your intersting argument.So this question gives an example of an infinite dimensional sphere with two complemented dense set each of them path connected..I doubt this could be hold in finite dimensional case
befor I read the details, in the linked paper the set $|x|\leq 1$ is called sphere but the usual terminology is ball or disk I think. any way in my question I mean the sphere the point of unit norm not less than 1. But I guess your argument still work, yes?
@DavidGao This is the reason that I did not include the Note part into the main question. If I have an obvious measure then I would ask what can be said about the measure of A or its complement
