I'm trying to prove or find a counterexample to the following: If $A$ is an uncountable collection of uncountable subsets $S$ of $\mathbb{R}$ such that each $x\in\mathbb{R}$ is in uncountably many of the $S\in A$, does it necessarily follow that there exists an intersection of uncountably many $S$ such that this intersection has an uncountable cardinality?
I'm really not sure where to start although I think transfinite induction might help. I know that there are definitely some positive examples (Like if $A$ is the set of all bounded intervals of $\mathbb{R}$) but obviously that doesn't cover all cases. I also tried to investigate a possibly easier case, namely infinite subsets of $\mathbb{N}$ which have a countably infinite intersection, but this seems just as hard.
You could consider the collection of all lines in $\mathbb{R}^2$.