# The Frame of Nuclei of an Alexandroff Space

## TL;DR

This paper characterizes when the frame of nuclei of the open set frame of an Alexandroff space is spatial, linking it to the embeddability of an infinite binary tree into the space's specialization preorder.

## Contribution

It establishes a precise condition involving the embedding of an infinite binary tree for the spatiality of the frame of nuclei in Alexandroff spaces.

## Key findings

- The frame of nuclei is spatial iff the infinite binary tree does not embed into the space.
- Provides a characterization connecting topological and order-theoretic properties.
- Bridges the structure of Alexandroff spaces with combinatorial tree properties.

## 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 of $\mathcal{O}S$. 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 $(S, \le)$, where $\le$ is the specialization preorder of $S$.