# When is the frame of nuclei spatial: A new approach

## TL;DR

This paper investigates the conditions under which the frame of nuclei is spatial by analyzing the density and scatteredness of nuclear points in the Esakia space, providing new characterizations and connecting to classical results.

## Contribution

It introduces a new approach to determine when the frame of nuclei is spatial by examining nuclear points in the Esakia space, unifying and extending previous results.

## Key findings

- L is spatial iff Y_L is dense in X_L
- N(L) is spatial iff Y_L is weakly scattered when L is spatial
- N(L) is boolean iff Y_L is scattered when L is spatial

## Abstract

For a frame $L$, let $X_L$ be the Esakia space of $L$. We identify a special subset $Y_L$ of $X_L$ consisting of nuclear points of $X_L$, and prove the following results: $L$ is spatial iff $Y_L$ is dense in $X_L$. If $L$ is spatial, then $N(L)$ is spatial iff $Y_L$ is weakly scattered. If $L$ is spatial, then $N(L)$ is boolean iff $Y_L$ is scattered. As a consequence, we derive the well-known results of Beazer and Macnab [1979], Simmons [1980], Niefield and Rosenthal [1987], and Isbell [1972].