# Are points in a variety always closed?

A point $P$ in a variety $X$ (affine,quasi-affine,projetive,quasi-projective)is closed if the closure $\overline{\{P\}}=\{P\}$.Will someone be kind enough to give me some hints on this?Thank you very much!

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Do you know the definition of the Zariski topology? If you understand the definition, then you shouldn't have any problems with the question. If you don't, then that's what you should be asking about. – Alex B. Aug 12 '11 at 15:04
Thank you for reminding!They are closed. – user14242 Aug 12 '11 at 15:19
@user14242: From the string of questions you have been asking, I'm beginning to get the impression you're trying to run before you can stand, to stretch a metaphor. Anyway. Your question is not clear. What do you mean by a variety—a locally ringed space in the sense of Hartshorne Chapter I, or an integral scheme of finite type over $\operatorname{Spec} k$, for some algebraically closed $k$? – Zhen Lin Aug 12 '11 at 15:22