In my textbook Introduction to Set Theory by Hrbacek and Jech, the authors first construct Cantor set:
Next they prove The relative complement of the Cantor set in $[0,1]$ is dense in $[0,1]$:
My question: I can not understand why the authors conclude
The open interval $\left(\frac{3k+1}{3^{n+1}}\,,\, \frac{3k+2}{3^{n+1}}\right)$ is certainly disjoint from $F$.
Could you please elaborate on this statement?
Thank you so much!
We remove the middle third of the interval to define the next $F_n$ and so it will be disjoint from $F_{n+1}$ so a fortiori from $F$.
Explanation upon request: An interval from $F_n$ is of the form $[\frac{k}{3^n}, \frac{k+1}{3^n}]$ (not all such intervals are in $F_n$ but $2^n$ of them are; this makes the exact formula for $F-n$ tricky: specify which $k$ do occur; hence the recursive definition with sequences etc.) and we can also write this as $[\frac{3k}{3^{n+1}}, \frac{3(k+1)}{3^{n+1}}] = [\frac{3k}{3^{n+1}}, \frac{3k+3)}{3^{n+1}}]$, multiplying both parts of the fraction by $3$ and so its middle third open interval is $(\frac{3k+1}{3^{n+1}}, \frac{3k+2)}{3^{n+1}})$, exactly as claimed.