Let $A$ be an uncountable set. Let $B \subseteq A$ be an uncountable proper subset of $A$. Is it true that $A-B$ is at most countable?
I think I can come up with a counter example: Let $B, C$ be uncountable sets such that $B \cap C = \emptyset$. Then let $A = B \cup C$. Then $A - B= C$ is uncountable.
However, I am trying prove something about the cocountable topology on an uncountable set $X$. I am trying to describe its open sets but I am not sure how to find subsets of $X$ such that its complement is at most countable.