Whether |R|=aleph_1 is independent of ZFC (but of course there are bijections both between R and R^2 and between aleph_1 and aleph_1^2, or between X and X^2 for any infinite set)
https://mathoverflow.net/questions/240249/continuous-map-fro...
https://mathoverflow.net/questions/34232/injective-maps-math...
The second link says that there's a continuous surjection from R -> R^2, but no such bijection.
(Which, after some googling looks in line with what I was imagining.)
If there's no bijection, how can the cardinalities be the same?
If there are holes in the map does it really cover the whole space?
As a simpler example: The map that multiplies each rational number by -1 and keeps irrational numbers as they are is bijective (it’s its own inverse), however it is nowhere continuous.
It is true that there is no continuous bijection R -> R^2. It is also true that this has nothing to with cardinality, which is only about the existence of bijections (and indeed is defined for arbitrary sets with no topological structure for which continuity makes no sense)
> Represent each real number x∈[0,1) as a sequence of DIGITS, where each DIGIT is either in {0,…,8} or is of the form 10∗(10k−1)+i with k≥1 and i∈{0,…,8} (i.e., in {90,…,98;990,…,998;9990,…,9998;…}.
He's iterating to infinity, that sure as shit sounds countable to me.
Anyway, the conclusion that I've come to is that you can map R onto a subset of R^n such that for all values {r[1] ... r[n]} in R^n there is a value in the R -> R^n mapping within all finite bounds.
So as long as you stay away from infinitesimals (not all of them), the domains are equivalent.
https://en.wikipedia.org/wiki/Infinitesimal
> In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero.
But as far as I can tell, the reverse mapping has an R^(n - 1) orthogonal that is completely unaccounted for, within those finite bounds.
(And breaking infinitesimals might cause some trouble for calculus.)
prefix = '0.';
queue = new Queue(prefix);
for(;;){
s = queue.poll();
for(c:['0'...'9']){
queue.add(s + c);
}
}
I think his iterator is generating infinitesimals of rational and not continuous nature.