The frontier of a set

I’m trying to prove that a certain set is dense in a metric space.
I have a metric space X, an open subspace $Ysubset X$ (same metric) and an open set $Usubset Y$ s.t $U$ is open and dense in $Y$. Moreover, X is Baire. I define $B=Xsetminus overline Y$ and tried proving that $Ucup B$ is dense in X.
Is the following proof correct?

$cl(Ucup B)=cl(U_ncup(Xsetminus cl(Y))=cl(U_n)cup cl(Xsetminus cl(Y))$

Now I claim that: $cl(Xsetminus cl(Y))=(Xsetminus cl(Y))cup Fr(Xsetminus cl(Y))=cl(Xsetminus cl(Y))=(Xsetminus (Ycup Fr(Y))cup (Fr(cl(Y)))=(Xsetminus Y)$

and therefore $cl(U_n)cup cl(Xsetminus cl(Y))=Ycup(Xsetminus Y)=X$

Is it true that $Fr(Y)=Fr(cl(Y))$ – specifically in a metric Baire space?