Show that the Cantor set, C, is totally disconnected.
Let $x,y\in C$ and suppose (WLOG) that $x\frac{1}{3^N}$ (where $\frac{1}{3^N}$ is the length of a closed interval in $C_N$. This implies that $x$ and $y$ belong to different closed intervals in $C_N$ and also, there must exist an element, $z$, where $xz, b\in C$}. Then $x\in A$ and $y\in B$. Note that $\bar A\cup B$=$\emptyset$ and $A\cup \bar B$=$\emptyset$, implying that $A$ and $B$ are separated sets. Then, since $A$ is the set of all elements in $C$ less than $z$ and $B$ is the set of all elements in $C$ greater than $z$ (and $z$ is not in $C$), it follows that $C=A\cup B$, and therefore $C$ is totally disconnected.
Is this correctly done? I am concerned about the part where I stated that $A$ and $B$ are separated sets, do I need to prove this point further?
Feedback is appreciated.
Thanks.
real-analysis
general-topology