Let $X$ be any topological space. If $x, y\in X$. We write $x\rightsquigarrow y$ if $y \in cl_X(x)$, where $cl_X(x)$ is the closure of $\{x\}$ in $X$, and we say $y$ is a specialization of $x$ or $x$ is a generalization of $y$. We write $min(X)$ for the elements of $X$, which are minimal w.r.t.$\rightsquigarrow$, and $max(X)$ for the elements of $X$, which are maximal w.r.t.$\rightsquigarrow$. I think the following facts are true:
1) $max(X)=$is the set of elements $x$ of $X$ such that for every $y\neq x$ in $X$ there exists an open set $U_y$ with $x\in U_y$ but $y\not\in U_y$.
2) $min(X)$ is the set of all closed points in $X$.
I am also looking for a good and simple refrenece for spectral space? Thanks for any help.