Let $\mathscr K$ be an uncountable set such that every $K\in\mathscr K$ is a compact subset of $[0,1]$ with positive Lebesgue measure. Does it then follow that there exists an uncountable $\mathscr A\subseteq\mathscr K$ with $\bigcap\,\mathscr A\not=\emptyset$ ?



Intersection of compact sets in the unit interval
Assuming the continuum hypothesis, the answer is negative.


First of all, we can index ${\cal K}$ by $[0,1]$, let $K\_x$ be the set in ${\cal K}$ with index $x$. Now well-order $[0,1]$ in a way corresponding to the first uncountable ordinal. For any $x$, there are only countably many points preceding $x$ in the well ordering. We can therefore choose $K\_x$ so that it does not contain any point that precedes $x$. Now let $A$ be any uncountable set and consider $\bigcap\_{x\in A} K\_x$. For any $y$, there is an $x\in A$ which comes after $y$. Hence this set must be empty.


