Say we have two algebraic sets, $X = V_{\mathbb A^2}(x+y-1)$ and $Y = V_{\mathbb A^2}(x^3+x^2y + y - 1)$.
Is it necessarily true that $\overline{X \cap Y} = \overline{X} \cap \overline{Y}$ for $\overline{X}$ indicating the closure of $X$ in $\mathbb P^2$? In general, is it true that for two algebraic sets is $\overline{X \cap Y} = \overline{X} \cap \overline{Y}$?
I'm also looking at how to determine the projective closure of algebraic sets, as I don't quite understand how to do so.