Proving a relationship between the closure and interior

Question

Let $X$ be a topological set. I want to prove that $X \backslash \overline{Y} = (X \backslash Y )^◦$.

We know that

$(X \backslash Y )^◦\subseteq X \backslash Y$ and $X \backslash \overline{Y} \subseteq X \backslash Y$, but I'm not sure how to use these.

Answer 1

If $x \in X\setminus \overline {Y}$ then $X\setminus \overline {Y}$ is an open set containing $x$ and contained in $x \in X\setminus {Y}$ hence $x$ is an interior point of $x \in X\setminus {Y}$. Converselly, suppose $x$ is an interior point of $x \in X\setminus {Y}$. We have to show that $x \notin \overline Y$. Since $(X \setminus Y)^{0}$ is an open set containing $x$ which contains no point of $Y$ it follows that $x \notin \overline Y$.

Answer 2

x not in $\overline Y$
iff some open U with x in U and empty U $\cap$ Y
iff some open U with x in U and U $\subseteq$ X - Y
iff x in $(X - Y)^o.$$\cap$