The question i'm stuck on right now is:

For an arbitrary $S \subseteq \mathbb{R}$, show that $(0,1) \cup S$ is equinumerous with $\mathbb{R}$.

My idea was to break this up into two cases, S being countable, and S being uncountable and see what happens. If S is countable, I can pictorially see the bijection, but I have no idea how to write it in an explicit form, and not sure how to cover the uncountable case.

  • 2
    $\begingroup$ Can you use Schröder-Bernstein? $\endgroup$ – Bungo Mar 6 '17 at 3:08
  • $\begingroup$ Yes, that would be a good idea. So I would need to find injective functions between the two sets then, correct? If so, not too sure how to go about this.. $\endgroup$ – user352879 Mar 6 '17 at 3:48
  • $\begingroup$ I bet you can find an injection from $(0,1)\cup S$ into $\mathbb R$ (what's the easiest map you can think of?) For the other direction, do you know an injection from $\mathbb R$ into $(0,1)$? $\endgroup$ – Bungo Mar 6 '17 at 4:02
  • $\begingroup$ Cantor-Bernstein Theorem, a.k.a. Schroeder-Bernstein Theorem, a.k.a Cantor-Schroeder-Bernstein Theorem. $\endgroup$ – DanielWainfleet Mar 6 '17 at 4:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.