Splitting an infinite set into two equal infinite subsets.

[ice109]

why does it seem to me like this is verging on trying to make onto functions from a function onto it's powerset? even before mathboy mentioned the p(r)

 

[Hurkyl]

P(A + B) and P(A) x P(B) are naturally bijective sets, right? So all you need to do is find sets A and B, and bijections

A + B ~ R
A ~ R
B ~ R

And voilà! You have a bijection P(R) ~ P(R) x P(R).

 

[andytoh]

Bijection F:P(R) x P(R) -> P(R):

Start with two subsets S and T of R. Get the characteristic functions (chi_S, chi_T). Get the characteristic function f:R -> {0,1} defined by

f(x)= (chi_S)(lnx) if x>0, (chi_T)(ln(-x)) if x<0. (problems with x=0, so someone find a better bijection from Rx{1,2} to R)

From the characteristic function f, get your unique subset of R corresponding to f. This subset of R is our F(S,T),
i.e. F(S,T)= {x|f(x)=1}.

Given S from P(R), let
A_S = F(P(R)x{S}).

The collection {A_S| S in P(R)} is the desired partition of P(R). All the A_S are pairwise disjoint, |A_S|=|P(R)|, and P(R) = U (A_S) .

Can someone find a better bijection from Rx{1,2} to R ?

 

[andytoh]

f:R->{0,1}

f(x)=
(chi_S)(ln(x+1)) if x is a non-negative integer
(chi_S)(lnx) if x>0 and not and integer,
(chi_T)(ln(-x)) if x<0.

Yuck. Can someone think of a better one?

 

[Hurkyl]

What's so yuck about it?

Anyways, observe the following: if you split the real line up into two or more intervals, at least one of them must be either a closed or a half-open interval. Such intervals cannot be homeomorphic to R.

Your method looks like one of the standard tricks for creating a point out of "thin air", so I don't see why it's so yucky.


Anyways, thinking about yours did lead me to a different, cute example: split the real line into half open intervals
... + [-2, -1) + [-1, 0) + [0, 1) + [1, 2) + [2, 3) + ...
then do your favorite rearrangement of Z into two copies of Z.