Show that $(0,1)\cup S$ is equinumerous with $\mathbb R$.

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.

• Can you use Schröder-Bernstein? – Bungo Mar 6 '17 at 3:08
• 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.. – user352879 Mar 6 '17 at 3:48
• 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 $(0,1)$? – Bungo Mar 6 '17 at 4:02
• Cantor-Bernstein Theorem, a.k.a. Schroeder-Bernstein Theorem, a.k.a Cantor-Schroeder-Bernstein Theorem. – DanielWainfleet Mar 6 '17 at 4:43