I encounter the definiton of generic point in Hartshorne's as follows:
A generic point for an irreducible closed subset $Z$ is a point $P$ such that $Z=${$\overline{P}$}, where {$\overline{P}$} denotes the closure of the set consisting of the set consisting of the point $P$.
I can't quite understand the meaning of "the set consisting of the point $P$". Does it mean the set consists only one point $P$? I have searched some examples. In a spectrum of a ring, the closure of a point, which is a prime ideal, is the set of all the prime ideals that contain the point. So in this case, the definition makes sense to me. Are there any different examples can explain the definition? Thanks!