Let $T$ be an uncountable set and $S$ ($S\subset T$) be a countable infinite subset of $T$. What can be said about $T \backslash S$? Is it countable or uncountable? Or do we need more information? At what point a subset of an uncountable set becomes infinite countable?
Complement of a countable infinite set in an uncountable set
elementary-set-theory
Best Answer
The union of two countable sets is countable (as is the union of countably many countable sets) so if $S \cup T$ is uncountable and $S$ is countable, then $T$ must be uncountable.