I'm trying to prove this equivalence which are also the definition of quasi-projective algebraic sets:
$X\subset \mathbb P^n$ is an open subset of its closure $\Leftrightarrow$ $X\subset \mathbb P^n$ is an open subset of a closed subset of $\mathbb P^n$.
The first side of this equivalence is easy, I need help with the converse.
I have another question:
The quasi-projective algebraic sets in Hartshorne's book are the open subsets of $\mathbb P^n$, I don't know why the definition above is equivalent to the definition in Hartshorne.
I really need help.
I would be very grateful if someone could help me
Thanks a lot.