Let $X$ be an uncountable set and consider a map $x\mapsto C_x$ assigning to every $x\in X$ a countable subset $C_x$ of $X$. Now for any $x\in X$ define $$R_x:=\{x'\in X|x\in C_{x'}\}.$$ Obviously $R_x$ can be any subset of $X$. It is also clear that $R_x=X$ can hold for at most countably many $x$. But what happens if $R_x$ is just a bit smaller than $X$? Specifically, is it possible that $R_x$ is co-countable for all $x$?
My intuition says NO. It is based on the following picture. For each $x$ $C_x$ can be taken to define a column in $X^2$: $$C_x':=\{(x,y)\in X^2|y\in C_x\}.$$ The union of all these columns defines a very "sparse" subset of $X^2$. The union can also be written as the union of all rows. If these rows would all be co-countable, the set doesn't look that "sparse" anymore. However, making this precise by turning $X^2$ into a measure space doesn't seem the way to go as the sets $C_x$ can be very wild.
Let $X$ be the set of countable ordinals, and let $C_x=\{w\in X\mid w \le x\}$.
Then for all $x\in X,\;C_x$ is countable, but $R_x$ is co-countable.