# How can a projective variety be regarded as a quasi-projective variety?

It can be seen that via projective closure, affine varieties, quasi-affine varieties, and projective varieties are all quasi-projective varieties .

How can I know that a projective variety is a quasi-projective variety via projective closure ？

• Topologically, a set $X$ is always open in itself. So if $X$ is a projective variety, then $X$ is an open subset (Zariski) of itself, so it is an open subset of a projective variety, and hence quasiprojective.
– Dave
Nov 30, 2023 at 16:34
• Indeed it is, but is that the correct interpretation? Nov 30, 2023 at 16:39
• Is it not possible to create a projective variety larger than the projective variety and become an open set in it, or something like that? Nov 30, 2023 at 16:49
• Generally, that is not possible. The image of a projective variety under a regular map is closed, so for a map of projective varieties to be an open embedding it has to be the inclusion of a connected component. Nov 30, 2023 at 17:00
• I understand that it is impossible. Thank you very much. Nov 30, 2023 at 17:18