Proof with Baire category theorem that the set $(0,1)$

Question

i am searching for the proof that the interval $(0,1)$ is not countable by using the Baire category theorem. Does someone know a book or has a reference for the proof ?

Thanks in advance.

Answer 1

You refer to this paper, page $7,$ paragraph after the proof of Theorem $3.2.$

Particularly, the proof goes as follows:

Suppose that $[0,1]$ is countable, and denote $$[0,1] =\bigcup_{n=1}^\infty r_n$$ Note that $[0,1]\setminus \{r_n\}$ is open dense set in $[0,1].$ By Baire category Theorem, the intersection $\bigcap_{n=1}^\infty [0,1]\setminus\{r_n\}$ is dense in $[0,1],$ that is, $$\bigcap_{n=1}^\infty [0,1]\setminus\{r_n\}\neq\emptyset.$$ However, $$\bigcap_{n=1}^\infty [0,1]\setminus\{r_n\}=\emptyset,$$ a contradiction.