My problem:
Suppose $E \subset [0,1]$ and $C \subset [0,1]$ are two compact sets such that $E \subsetneq C$ and $\mathop C\limits^ \circ =\mathop E\limits^ \circ =\emptyset$ and $|E|=|C|=0$.
Can I say that there exist $0 \leq a<b \leq 1$ such that $]a,b[ \subset [0,1]-C$ and $\{a,b\} \cap C-E \neq \emptyset$?
My attempt:
I tried writing $C^c$ as an opportune union of connected components, but I cannot reach any conclusion.
HINT:
Pick a point $x\in C-E $. $E$ is closed, therefore you can find $r>0$ so that $(x-r, x+r)\cap E = \emptyset$. Now look for suitable $a,b \in (x-r, x+r)\cap [0,1]$.
EDIT: adding further details.
Suppose $x\neq 1$ and define $A:=(x,x+r)$.
If $A\cap C = \emptyset$, a possible solution is $a=x,$ $b= \min\{x+r,1\}$.
Suppose now $y \in A\cap C$. The interval $(x,y)$ cannot be a subset of $C$, because $\mathop C\limits^ \circ = \emptyset$. For this reason, there exists $ z\in (x,y) - C$. You can now choose $(a,b)$ as the connected component of $z$ in $C^c$.
If $x=1$, you can look at $B:=(x-r,x)$ instead of $A$ and exploit the same idea.