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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.

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Equivalences have directions, not sides, so I'm not sure which one you're asking about. Is it "$\Leftarrow$"?

If $X$ is open in $Y$, which is closed in $\mathbf{P}^n$, then $X$ is open in $Y\cap \overline{X}$. As $Y$ is closed and contains $X$, $Y$ contains $\overline{X}$, so $Y\cap \overline{X} = \overline{X}$

I don't know which Hartshorne book you're looking at, but the one I'm looking at defines on p. 10 a quasi-projective variety as an open subset of a projective variety, not projective space.

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