Lonely points in $\omega^*$

TL;DR

This paper introduces the concept of lonely points in the Stone-Čech remainder of the natural numbers, linking their existence to specific topological spaces and exploring methods to find them within *.

Contribution

It defines lonely points, establishes their equivalence to certain topological space properties, and analyzes methods for constructing them in *.

Findings

Existence of lonely points is equivalent to a specific topological space condition.

Methods can find lonely points in large subspaces of *.

Known methods cannot construct lonely points in all of *.

Abstract

Inspired by the work of J. van Mill we define a new topological type --- lonely points. We show that the question of whether these points exist in $\omega^*$ is equivalent to finding a countable OHI, extremally disconnected, zerodimensional space with a remote weak P-point. We also present methods which allow us to find lonely points in a large subspace of $\omega^*$ and show why known methods do not allow us to construct them in all of $\omega^*$.