Subset of an uncountable set

Question

For all uncountable set,

Is there an uncountable subset such that its complement is also uncountable?

How can I prove this?

@Elaqqad Infinite does not imply uncountable. – Tim Raczkowski Apr 24 '15 at 14:30 — Tim Raczkowski, Apr 24 '15 at 14:30

score 1 · Answer 1 · answered Apr 24 '15 at 14:26

1

Assuming AC the answer is yes. (I'm not sure what the answer is in ZF).

Hint: Think about even and odd ordinals, and how you might use their existence for your purposes. This actually proves a bit more namely that you can split any infinite set into two sets with cardinality equal to the original set.

answered Apr 24 '15 at 14:26

DRF

5,282

what's meant by AC and ZF? – Mirumommy Apr 24 '15 at 14:30
3

Without the axiom of choice it is consistent that the answer is negative. – Asaf Karagila Apr 24 '15 at 14:32
@asaf I figured you could probably construct some annoying uncountable set with that property. Though I suppose you have try a bit since I guess you can't do it for the reals as the other answer helpfully points out? – DRF Apr 24 '15 at 14:34
Been there, done that. The actual construction is a bit eeky. I can get into details if you want, I may have already done that, too. – Asaf Karagila Apr 24 '15 at 14:38
Thanks. But I don't know even and odd ordinals. – Evzone Apr 24 '15 at 19:01

Subset of an uncountable set

1 Answers1