# Does the closure of component set restricted to subspace equals to the closure of component set in the subspace?

Let $X$ is a topological space, $Y$ is a subspace of $X$, $A \subseteq Y$. Then I know $$Y\cap Cl_X(A)=Cl_Y(A)$$ holds.

But does $$Y\cap Cl_X(X-A)=Cl_Y(Y-A)$$ also holds?

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So your question is $Cl_Y(Y-A)=Cl_Y(X-A)$? –  Siminore Sep 14 '12 at 15:02
@Siminore I think not, because $X-A$ may not included in $Y$. –  Popopo Sep 14 '12 at 15:32

Take for example $X = \mathbb R, Y = [0, 2]$ and $A = [0, 1]$. Then $0 \in Y \cap Cl_X(X - A)$ but $0 \not \in Cl_Y(Y - A)$.
So that means $Int_Y(A)=Y \cap Int_X(A)$ is also not true in general. –  Popopo Sep 15 '12 at 4:17