removing a countable subset from an infinite set doesn't change cardinality

Question

Prove that if an infinite set $S$ has a countable (infinite or finite) subset removed from it leaving the set $T$, and $T$ is infinite then: $$|S|=|T|$$

I have seen some proofs related to this but usually only regarding removing a finite set.

Any help would be appreciated.

Answer 1

Hint: let be $R$ the countably infinite subset s.t. $S = T\cup R$. Extract $Q\subset T$ countably infinite. Construct a bijection between $Q$ and $R\cup Q$.

Answer 2

If S is countable and T is infinite, then T must be countable. If S is uncountably infinite, then removing a countable set leaves a noncountable set behind.