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?
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?
The assertion is false.
Take for example $X = \mathbb R, Y = [0, 2]$ and $A = [0, 1]$. Then $0 \in Y \cap Cl_X(X - A)$ but $0 \not \in Cl_Y(Y - A)$.