On points of convergence lattices and sobriety for convergence spaces

TL;DR

This paper characterizes certain convergence spaces through their point spaces in convergence lattices, explores variants of sobriety and $T_D$ axioms, and connects these concepts to sobrification, revealing new phenomena beyond topology.

Contribution

It provides a characterization of convergence spaces via their point spaces in convergence lattices and investigates sobriety variants and their relation to sobrification.

Findings

Characterization of convergence spaces via point spaces in convergence lattices.

Introduction of new sobriety variants and $T_D$ axiom studies in convergence spaces.

Identification of a quotient homeomorphic to sobrification for certain convergence lattices.

Abstract

We characterize the convergence spaces $(X,\xi)$ such that the space of points of $(\mathbb{P}X,\lim_{\xi})$ in the category of convergence lattices is $(X,\xi)$. On the way, we study variants of sobriety and of the axiom $T_{D}$ in convergence spaces. New phenomena appear when leaving the realm of topological spaces. We obtain new hindsight into the space of points of a convergence lattice and study a special quotient of it, which, in the case $L=(\mathbb{P}X,\lim_{\xi})$ for a topological space $(X,\xi)$, turns out to be homeomorphic to the sobrification of $X$.