I am following Ravi Vakil's Math 216: Foundations of Algebraic geometry notes, and there is a remark following an exercise that I don't understand at all, and if anyone could enlighten me then that would be brilliant.
The exercise asks one to show that if $X$ is a quasicompact scheme, then every point has a closed point in its closure, which is clear from the preceding exercise asking to show that $X$ is quasicompact if and only if it can be written as a finite union of affine schemes. These I am fine, as well as the following implication that every nonempty closed subset of $X$ contains a closed point.
However, then the notes then go on to state that this will be used in the following way: If a property $P$ is open (that is, if some point $x$ has $P$, then there exists an open neighbourhood $U$ of $x$ such that all points in $U$ have $P$), then to check that all points of a quasicompact scheme have $P$, then it suffices to check only the closed points.
I do not seem to be able to see how this follows at all. It seems to me that everything in the exercises is regarding closed points being in closures, and to show the remark, I want to show that other points are in (all) open neighbourhood(s) if closed points. These seem relatively distinct to me - is this wrong?
These comments/exercises are on pages 139-140 of the notes.