How can some infinities be bigger than others?

Re:How can some infinities be bigger than others? by Skeime on Thursday April 20, 2023 @04:55AM (#35637741)

You can get from R^d to I^d (with I the open unit interval) easily by applying x -> (arctan(x) + pi/2)/pi and y -> tan(pi * y - pi/2) to each coordinate. There are also ways to go from R to the closed unit interval. You can then combine this with space-filling curves for I^d.

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Re:How can some infinities be bigger than others? by yarg on Thursday April 20, 2023 @05:09AM (#35637805)

You can even go from R to I^d.

The problem that I'm having is in getting from R to R^d, which I still don't think is possible - look at the comment where I mention "surjections" and "bijections".

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Re:How can some infinities be bigger than others? by Skeime on Thursday April 20, 2023 @01:10PM (#35643030)

By applying the map from I^d to R^d (using the tan-arctan-construction above in each coordinate) after the map from R to R^d, you get a map from R to R^d.

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Re:How can some infinities be bigger than others? by yarg on Thursday April 20, 2023 @10:42PM (#35649210)

I^n is just a countably infinite subset of R^n.

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Re:How can some infinities be bigger than others? by Skeime on Saturday April 22, 2023 @07:44AM (#35665549)

No, I^n is also uncountable. In fact, the usual proof for the uncountability of R is really proof of the uncountability of I, and then the natural injection into R is used to extend it to R.

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