Let $\displaystyle Y\subset X$ and let $\displaystyle A\subset X$. If $\displaystyle A$ is countable, $\displaystyle X$ is complete, and $\displaystyle Y\cup A=X$, is $\displaystyle Y$ necessarily dense in $\displaystyle X$?

Prove it or give a counterexample.

If the answer is no, then does making $\displaystyle X$ compact change anything?