The Frame of Nuclei on an Alexandroff Space

Abstract

Let $\mathcal {O} S$ be the frame of open sets of a topological space S, and let $N(\mathcal {O} S)$ be the frame of nuclei on $\mathcal {O} S$ . For an Alexandroff space S, we prove that $N(\mathcal {O} S)$ is spatial iff the infinite binary tree ${\mathscr{T}}_2$ does not embed isomorphically into (S,≤), where ≤ is the specialization preorder of S.